\hSrSSKJr SSKJr SSKJr SSKJr SSK J r J r J r J r SSKJr SSKJr SS KJr SS KJrJrJr SS KJr SS KJr SS KJrJrJrJ r J!r!J"r" SSK#J$r$J%r%J&r&J'r' SSK(J)r)J*r*J+r+J,r,J-r-J.r. SSK/J0r0J1r1J2r2 SSK3J4r4 SSK5J6r6 SSK7J8r8J9r9J:r:J;r;Jr> SSK?J@r@JArA SSKBJCrCJDrD SSKEJFrFJGrGJHrH SSKIJJrJ SSKKJLrL SSKMJNrN SSKOJPrPJQrQJRrR SSKSJTrT SSKUJVrV SSKWJXrXJYrY SSKZJ[r\ S r]\ "S!5r^"S"S#\5r_"S$S%\5r`"S&S'\5ra"S(S)\5rb"S*S+\5rc"S,S-\5rd"S.S/\5re"S0S1\5rf"S2S3\5rg"S4S5\5rh"S6S7\5ri"S8S9\5rj"S:S;\5rk"S<S=\5rl"S>S?\5rm"S@SA\5rn"SBSC\5ro"SDSE\5rp"SFSG\5rq"SHSI\5rr"SJSK\5rs"SLSM\5rt"SNSO\5ru"SPSQ\v5rwSRrxSSry\z"ST\y5r{SUr|SeSVjr}\SeSWj5r~\SX5rSeSYjrSZr\S[5rS\r\S]5rSfS^jr\S_5rSgS`jr"SaSb\5rShSTSTSc.SdjjrgT)ia6 This module implements some special functions that commonly appear in combinatorial contexts (e.g. in power series); in particular, sequences of rational numbers such as Bernoulli and Fibonacci numbers. Factorials, binomial coefficients and related functions are located in the separate 'factorials' module. ) annotations)prod) defaultdict)Callable)SSymbolAddDummy)cacheit)Dict)Expr)ArgumentIndexErrorDefinedFunction expand_mul) fuzzy_not)Mul)EIpiooRationalInteger)Eqis_leis_gtis_lt) SYMPY_INTSremovelcmlegendrejacobi kronecker)binomial factorial subfactoriallog) Piecewise) factorint_divisor_sigma is_carmichael find_carmichael_numbers_in_rangefind_first_n_carmichaels)_primepi) _partition_partition_rec)isprime is_square)bernoulli_poly euler_poly genocchi_poly)cancel)MultisetPartitionTraversersympy_deprecation_warning)multisetmultiset_derangementsiterable)recurrence_memo)as_int)mpworkprec)ifibc0[[XS-55$N)rrange)abs ]/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/combinatorial/numbers.py_productrI-s aQ  xcx\rSrSrSr\S5r\S5r\S5r\S5r \S5r \S5r S r g ) carmichael;a Carmichael Numbers: Certain cryptographic algorithms make use of big prime numbers. However, checking whether a big number is prime is not so easy. Randomized prime number checking tests exist that offer a high degree of confidence of accurate determination at low cost, such as the Fermat test. Let 'a' be a random number between $2$ and $n - 1$, where $n$ is the number whose primality we are testing. Then, $n$ is probably prime if it satisfies the modular arithmetic congruence relation: .. math :: a^{n-1} = 1 \pmod{n} (where mod refers to the modulo operation) If a number passes the Fermat test several times, then it is prime with a high probability. Unfortunately, certain composite numbers (non-primes) still pass the Fermat test with every number smaller than themselves. These numbers are called Carmichael numbers. A Carmichael number will pass a Fermat primality test to every base $b$ relatively prime to the number, even though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie-PSW primality test and the Miller-Rabin primality test. Examples ======== >>> from sympy.ntheory.factor_ import find_first_n_carmichaels, find_carmichael_numbers_in_range >>> find_first_n_carmichaels(5) [561, 1105, 1729, 2465, 2821] >>> find_carmichael_numbers_in_range(0, 562) [561] >>> find_carmichael_numbers_in_range(0,1000) [561] >>> find_carmichael_numbers_in_range(0,2000) [561, 1105, 1729] References ========== .. [1] https://en.wikipedia.org/wiki/Carmichael_number .. [2] https://en.wikipedia.org/wiki/Fermat_primality_test .. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents c.[SSSS9 [U5$)Nzt is_perfect_square is just a wrapper around sympy.ntheory.primetest.is_square so use that directly instead. 1.11$deprecated-carmichael-static-methodsdeprecated_since_versionactive_deprecations_target)r9r2ns rHis_perfect_squarecarmichael.is_perfect_squarens#! "(#I  |rJc&[SSSS9 X-S:H$)NzI divides can be replaced by directly testing n % p == 0. rPrQrRrr8)prVs rHdividescarmichael.divideszs%! "(#I  uzrJc.[SSSS9 [U5$)Nzi is_prime is just a wrapper around sympy.ntheory.primetest.isprime so use that directly instead. rPrQrR)r9r1rUs rHis_primecarmichael.is_primes#! "(#I  qzrJc.[SSSS9 [U5$)Nzr is_carmichael is just a wrapper around sympy.ntheory.factor_.is_carmichael so use that directly instead. 1.13%deprecated-ntheory-symbolic-functionsrR)r9r+rUs rHr+carmichael.is_carmichaels$! "(#J  QrJc.[SSSS9 [X5$)Nz find_carmichael_numbers_in_range is just a wrapper around sympy.ntheory.factor_.find_carmichael_numbers_in_range so use that directly instead. rarbrR)r9r,)rKys rHr,+carmichael.find_carmichael_numbers_in_ranges$! "(#J  055rJc.[SSSS9 [U5$)Nz find_first_n_carmichaels is just a wrapper around sympy.ntheory.factor_.find_first_n_carmichaels so use that directly instead. rarbrR)r9r-rUs rHr-#carmichael.find_first_n_carmichaelss$! "(#J  (**rJN) __name__ __module__ __qualname____firstlineno____doc__ staticmethodrWr[r^r+r,r-__static_attributes__rirJrHrMrM;s0d         6 6 + +rJrMc\rSrSrSr\S5r\\"S\R\ /5S55r \ S Sj5r SrSrS rS rg) fibonacciaC Fibonacci numbers / Fibonacci polynomials The Fibonacci numbers are the integer sequence defined by the initial terms `F_0 = 0`, `F_1 = 1` and the two-term recurrence relation `F_n = F_{n-1} + F_{n-2}`. This definition extended to arbitrary real and complex arguments using the formula .. math :: F_z = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5} The Fibonacci polynomials are defined by `F_1(x) = 1`, `F_2(x) = x`, and `F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)` for `n > 2`. For all positive integers `n`, `F_n(1) = F_n`. * ``fibonacci(n)`` gives the `n^{th}` Fibonacci number, `F_n` * ``fibonacci(n, x)`` gives the `n^{th}` Fibonacci polynomial in `x`, `F_n(x)` Examples ======== >>> from sympy import fibonacci, Symbol >>> [fibonacci(x) for x in range(11)] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55] >>> fibonacci(5, Symbol('t')) t**4 + 3*t**2 + 1 See Also ======== bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci References ========== .. [1] https://en.wikipedia.org/wiki/Fibonacci_number .. [2] https://mathworld.wolfram.com/FibonacciNumber.html c[U5$N)_ifibrUs rH_fibfibonacci._fibs QxrJNcBUS[US--R5$)N_symexpandrVprevs rH_fibpolyfibonacci._fibpolys$R4R=(0022rJczU[RLa[R$UR(aUcN[U5nUS:a#[RUS--[ U*5-$[ URU55$US:a [S5eURU5R[U5$g)NrrDzDFibonacci polynomials are defined only for positive integer indices.) rInfinity is_Integerint NegativeOnerrrrw ValueErrorrsubsr}clsrVsyms rHevalfibonacci.evals  ?::  <<{Fq5==1q51IqbMAA"388A;//q5$&<==||A++D#66 rJc SSKJnJn [RU-U"[R U-5[RU-- - U"S5- $)Nr)sqrtcos)sympy.functionsrrr GoldenRatioPi)selfrVkwargsrrs rH_eval_rewrite_as_tractable$fibonacci._eval_rewrite_as_tractables<- q 3qttAv;q}}a/?#??aHHrJc pSSKJn SU*-U"S5-SU"S5-U-U"S5*S-U-- -S- $)NrrrrD(sympy.functions.elementary.miscellaneousrrrVrrs rH_eval_rewrite_as_sqrtfibonacci._eval_rewrite_as_sqrtsDAA2wtAwT!Wq 0T!WHqL13D DEIIrJc [RU-S[R*U-- - S[R-S- - $NrDr)rrrrVrs rH_eval_rewrite_as_GoldenRatio&fibonacci._eval_rewrite_as_GoldenRatio s8 q 1q}}nq%8#881Q]]?1;LMMrJriru)rjrkrlrmrnrorwr=rOner}r classmethodrrrrrprirJrHrrrrsn'RdAEE4()3*377"IJNrJrrc.\rSrSrSr\S5rSrSrg)lucasia} Lucas numbers Lucas numbers satisfy a recurrence relation similar to that of the Fibonacci sequence, in which each term is the sum of the preceding two. They are generated by choosing the initial values `L_0 = 2` and `L_1 = 1`. * ``lucas(n)`` gives the `n^{th}` Lucas number Examples ======== >>> from sympy import lucas >>> [lucas(x) for x in range(11)] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123] See Also ======== bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci References ========== .. [1] https://en.wikipedia.org/wiki/Lucas_number .. [2] https://mathworld.wolfram.com/LucasNumber.html cU[RLa[R$UR(a[US-5[US- 5-$grC)rrrrrrrVs rHr lucas.eval5s@  ?::  <<QU#iA&66 6 rJc XSSKJn SU*-SU"S5-U-U"S5*S-U---$)NrrrrDrrrs rHrlucas._eval_rewrite_as_sqrt=s7AA2wT!Wq(T!WHqL1+<<==rJriN) rjrkrlrmrnrrrrprirJrHrrs >77>rJrc\rSrSrSr\\"\R\R\R/5S55r \\"\R\R\ S-/5S55r \ S Sj5rSrS rS rg) tribonacciiIa@ Tribonacci numbers / Tribonacci polynomials The Tribonacci numbers are the integer sequence defined by the initial terms `T_0 = 0`, `T_1 = 1`, `T_2 = 1` and the three-term recurrence relation `T_n = T_{n-1} + T_{n-2} + T_{n-3}`. The Tribonacci polynomials are defined by `T_0(x) = 0`, `T_1(x) = 1`, `T_2(x) = x^2`, and `T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)` for `n > 2`. For all positive integers `n`, `T_n(1) = T_n`. * ``tribonacci(n)`` gives the `n^{th}` Tribonacci number, `T_n` * ``tribonacci(n, x)`` gives the `n^{th}` Tribonacci polynomial in `x`, `T_n(x)` Examples ======== >>> from sympy import tribonacci, Symbol >>> [tribonacci(x) for x in range(11)] [0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149] >>> tribonacci(5, Symbol('t')) t**8 + 3*t**5 + 3*t**2 See Also ======== bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition References ========== .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers .. [2] https://mathworld.wolfram.com/TribonacciNumber.html .. [3] https://oeis.org/A000073 c$USUS-US-$)Nrzr{rirs rH_tribtribonacci._tribps R48#d2h./rJrcbUS[US--[S-US--R5$)Nrrzrr{r|rs rH _tribpolytribonacci._tribpolyus6R4R=(4748+;;CCEErJNc(U[RLa[R$UR(a^[U5nUS:a [ S5eUc[ UR U55$URU5R[U5$g)NrzITribonacci polynomials are defined only for non-negative integer indices.) rrrrrrrrrr}rs rHrtribonacci.evalzs|  ?::  <<AA1u "@AA{syy|,,}}Q',,T377 rJc SSKJnJn S[RU"S5--S- nSU"SSU"S5--5-U"SSU"S5-- 5-S- nSXS"SSU"S5--5--US-U"SSU"S5-- 5--S- nSUS-U"SSU"S5--5--XS"SSU"S5-- 5--S- nXaS--Xg- Xh- -- XqS--Xv- Xx- -- -XS--X- X- -- -n U $) Nrcbrtrr{rrD!)rrrr ImaginaryUnit) rrVrrrwrFrGcTns rHr tribonacci._eval_rewrite_as_sqrts4G !//DG+ +q 0 b1T"X:o& &b1T"X:o)> >! C 4QtBxZ(( (1a4R!DH*_0E+E E J Ad2$r( ?++ +aR!DH*_0E.E E Ja%j1515/*a%j1515/*+a%j1515/*+ rJc SSKJn SSKJnJn U"SSU"S5--5nSU-[ R U--US-SU-- S -- nU"U[ R-5$) NrfloorriJfrrr)#sympy.functions.elementary.integersrrrrrTribonacciConstantHalf)rrVrrrrrGrs rH#_eval_rewrite_as_TribonacciConstant.tribonacci._eval_rewrite_as_TribonacciConstants`=G s48|# $ UQ))1, ,1qs Q ?R!&&[!!rJriru)rjrkrlrmrnror=rZerorrr}rrrrrrprirJrHrrIs$LaffaeeQUU+,0-0affaeeT1W-.F/F 8 8 "rJrc\rSrSr%SrS\S'\S5r\R\ "SS5\ "SS5\ "S S 5S .r S SSS S .r \ SSj5rSSjrSrSrg) bernoulliia* Bernoulli numbers / Bernoulli polynomials / Bernoulli function The Bernoulli numbers are a sequence of rational numbers defined by `B_0 = 1` and the recursive relation (`n > 0`): .. math :: n+1 = \sum_{k=0}^n \binom{n+1}{k} B_k They are also commonly defined by their exponential generating function, which is `\frac{x}{1 - e^{-x}}`. For odd indices > 1, the Bernoulli numbers are zero. The Bernoulli polynomials satisfy the analogous formula: .. math :: B_n(x) = \sum_{k=0}^n (-1)^k \binom{n}{k} B_k x^{n-k} Bernoulli numbers and Bernoulli polynomials are related as `B_n(1) = B_n`. The generalized Bernoulli function `\operatorname{B}(s, a)` is defined for any complex `s` and `a`, except where `a` is a nonpositive integer and `s` is not a nonnegative integer. It is an entire function of `s` for fixed `a`, related to the Hurwitz zeta function by .. math:: \operatorname{B}(s, a) = \begin{cases} -s \zeta(1-s, a) & s \ne 0 \\ 1 & s = 0 \end{cases} When `s` is a nonnegative integer this function reduces to the Bernoulli polynomials: `\operatorname{B}(n, x) = B_n(x)`. When `a` is omitted it is assumed to be 1, yielding the (ordinary) Bernoulli function which interpolates the Bernoulli numbers and is related to the Riemann zeta function. We compute Bernoulli numbers using Ramanujan's formula: .. math :: B_n = \frac{A(n) - S(n)}{\binom{n+3}{n}} where: .. math :: A(n) = \begin{cases} \frac{n+3}{3} & n \equiv 0\ \text{or}\ 2 \pmod{6} \\ -\frac{n+3}{6} & n \equiv 4 \pmod{6} \end{cases} and: .. math :: S(n) = \sum_{k=1}^{[n/6]} \binom{n+3}{n-6k} B_{n-6k} This formula is similar to the sum given in the definition, but cuts `\frac{2}{3}` of the terms. For Bernoulli polynomials, we use Appell sequences. For `n` a nonnegative integer and `s`, `a`, `x` arbitrary complex numbers, * ``bernoulli(n)`` gives the nth Bernoulli number, `B_n` * ``bernoulli(s)`` gives the Bernoulli function `\operatorname{B}(s)` * ``bernoulli(n, x)`` gives the nth Bernoulli polynomial in `x`, `B_n(x)` * ``bernoulli(s, a)`` gives the generalized Bernoulli function `\operatorname{B}(s, a)` .. versionchanged:: 1.12 ``bernoulli(1)`` gives `+\frac{1}{2}` instead of `-\frac{1}{2}`. This choice of value confers several theoretical advantages [5]_, including the extension to complex parameters described above which this function now implements. The previous behavior, defined only for nonnegative integers `n`, can be obtained with ``(-1)**n*bernoulli(n)``. Examples ======== >>> from sympy import bernoulli >>> from sympy.abc import x >>> [bernoulli(n) for n in range(11)] [1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66] >>> bernoulli(1000001) 0 >>> bernoulli(3, x) x**3 - 3*x**2/2 + x/2 See Also ======== andre, bell, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci, sympy.polys.appellseqs.bernoulli_poly References ========== .. [1] https://en.wikipedia.org/wiki/Bernoulli_number .. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial .. [3] https://mathworld.wolfram.com/BernoulliNumber.html .. [4] https://mathworld.wolfram.com/BernoulliPolynomial.html .. [5] Peter Luschny, "The Bernoulli Manifesto", https://luschny.de/math/zeta/The-Bernoulli-Manifesto.html .. [6] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 ztuple[Integer]argscSn[[US-US- 55n[SUS-S-5HUnX[USU-- 5-- nU[ US- SU-- S-USU-- 5-nU[ SU-S-SU-S-5-nMW US-S:Xa[ US-S5*U- nO[ US-S5U- nU[US-U5- $)NrrrDr )rr#rErrIr)rVsrFjs rH_calc_bernoullibernoulli._calc_bernoulli s  QA& 'q!Q$(#A Yq1Q3w'' 'A !a%!A#+/1qs73 3A (1Q37AaC!G, ,A $ q5A:!a%##a'AQ"Q&A8AE1%%%rJrDrrr{)rrDrrrrNcU[RLaU"U5$UR(a[R$URSLdURSLa6Ub2UR (a!UR (a[R$gUGc"U[RLa[R$UR(a$US- R(a[R$UR(a[U5nUS:a6[R"U5up4[![U5[U55$US-nUR"UnX::aUR$U$['US-US-S5H0nUR)U5nXR$U'XpR"U'M2 W$gUR(a [+X5$g)NFrDir)rris_zero is_integeris_nonnegativeris_nonpositiveNaNris_odd is_positiver is_Numberrr?bernfracr_highest_cacherErr3) rrVrKrZqcasehighest_cachedirGs rHrbernoulli.evalsc :q6M YY55L \\U "a&6&6%&?}!2B2Buu  YAEEzvv qs//vv Fs7;;q>DA#CFCF331u!$d!3&::a=(~11q5!(bernoulli._eval_evalf..J2 1;; rrD?)allr _to_mpmathlenrrr@r?mpfisintrbernpolyrr _from_mpmath)rprecrVrKress rH _eval_evalfbernoulli._eval_evalfIs2 222  IIaL # #D ) ^a/TYYq\QUU F Ft L d^AvffQia"&&+o!a)*abll1oR[[5Fb2771Q3?*  d++^s B0E"" E0rirurD)rjrkrlrmrn__annotations__rorrrrrrrrrrrprirJrHrrsxbH  & & 8Aq>hq!n"bAQ RFqQ'H#(#(JC,rJrc\rSrSrSr\\"SS/5S55r\\"\R\ /5S55r \S5r \ S Sj5rS S jrS rg) belliaa Bell numbers / Bell polynomials The Bell numbers satisfy `B_0 = 1` and .. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k. They are also given by: .. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}. The Bell polynomials are given by `B_0(x) = 1` and .. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x). The second kind of Bell polynomials (are sometimes called "partial" Bell polynomials or incomplete Bell polynomials) are defined as .. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) = \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n} \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!} \left(\frac{x_1}{1!} \right)^{j_1} \left(\frac{x_2}{2!} \right)^{j_2} \dotsb \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}. * ``bell(n)`` gives the `n^{th}` Bell number, `B_n`. * ``bell(n, x)`` gives the `n^{th}` Bell polynomial, `B_n(x)`. * ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind, `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`. Notes ===== Not to be confused with Bernoulli numbers and Bernoulli polynomials, which use the same notation. Examples ======== >>> from sympy import bell, Symbol, symbols >>> [bell(n) for n in range(11)] [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975] >>> bell(30) 846749014511809332450147 >>> bell(4, Symbol('t')) t**4 + 6*t**3 + 7*t**2 + t >>> bell(6, 2, symbols('x:6')[1:]) 6*x1*x5 + 15*x2*x4 + 10*x3**2 See Also ======== bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci References ========== .. [1] https://en.wikipedia.org/wiki/Bell_number .. [2] https://mathworld.wolfram.com/BellNumber.html .. [3] https://mathworld.wolfram.com/BellPolynomial.html rDcZSnSn[SU5HnX0U- -U-nX#X-- nM U$rC)rErVrrrFks rH_bell bell._bellsC  q!AU q A TW ArJcSnSn[SUS-5HnX0U- S--US- -nX#XS- -- nM! [[U-5$r)rErr}rs rH _bell_polybell._bell_polys]  q!a%AUQYAE*A Ta%[ A!$(##rJc^US:XaUS:Xa[R$US:XdUS:Xa[R$[Rn[Rn[SX- S-5H6nX4[R X- US- U5-X%S- -- nX@U- -U- nM8 [ U5$)aT The second kind of Bell polynomials (incomplete Bell polynomials). Calculated by recurrence formula: .. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) = \sum_{m=1}^{n-k+1} \x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k}) where `B_{0,0} = 1;` `B_{n,0} = 0; for n \ge 1` `B_{0,k} = 0; for k \ge 1` rrDr)rrrrEr _bell_incomplete_polyr)rVrsymbolsrrFms rHrbell._bell_incomplete_polys" Fa55L1f!q&66M FF EEq!%!)$A T//q1ug'')0Q8 8AU aA%!}rJNcU[RLaUc[R$[S5eUR(dURSLa [S5eUR (aUR (a~Uc#[UR[U555$Uc.UR[U55R[U5$UR[U5[U5U5nU$gg)NzBell polynomial is not definedFza non-negative integer expected)rrr is_negativerrrrrrrrr}r)rrVk_symrrs rHr bell.evals  ?}zz! !ABB ==ALLE1>? ? <@aVaeeT]#$$$8& DrJr c\rSrSr%Sr0rS\S'\SSj5r\ R4Sjr SSjr SS jr SS jr\ R4S jrS rSS jrSrSSjrSrg)harmonicia, Harmonic numbers The nth harmonic number is given by `\operatorname{H}_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}`. More generally: .. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m} As `n \rightarrow \infty`, `\operatorname{H}_{n,m} \rightarrow \zeta(m)`, the Riemann zeta function. * ``harmonic(n)`` gives the nth harmonic number, `\operatorname{H}_n` * ``harmonic(n, m)`` gives the nth generalized harmonic number of order `m`, `\operatorname{H}_{n,m}`, where ``harmonic(n) == harmonic(n, 1)`` This function can be extended to complex `n` and `m` where `n` is not a negative integer or `m` is a nonpositive integer as .. math:: \operatorname{H}_{n,m} = \begin{cases} \zeta(m) - \zeta(m, n+1) & m \ne 1 \\ \psi(n+1) + \gamma & m = 1 \end{cases} Examples ======== >>> from sympy import harmonic, oo >>> [harmonic(n) for n in range(6)] [0, 1, 3/2, 11/6, 25/12, 137/60] >>> [harmonic(n, 2) for n in range(6)] [0, 1, 5/4, 49/36, 205/144, 5269/3600] >>> harmonic(oo, 2) pi**2/6 >>> from sympy import Symbol, Sum >>> n = Symbol("n") >>> harmonic(n).rewrite(Sum) Sum(1/_k, (_k, 1, n)) We can evaluate harmonic numbers for all integral and positive rational arguments: >>> from sympy import S, expand_func, simplify >>> harmonic(8) 761/280 >>> harmonic(11) 83711/27720 >>> H = harmonic(1/S(3)) >>> H harmonic(1/3) >>> He = expand_func(H) >>> He -log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1)) + 3*Sum(1/(3*_k + 1), (_k, 0, 0)) >>> He.doit() -log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3 >>> H = harmonic(25/S(7)) >>> He = simplify(expand_func(H).doit()) >>> He log(sin(2*pi/7)**(2*cos(16*pi/7))/(14*sin(pi/7)**(2*cos(pi/7))*cos(pi/14)**(2*sin(pi/14)))) + pi*tan(pi/14)/2 + 30247/9900 >>> He.n(40) 1.983697455232980674869851942390639915940 >>> harmonic(25/S(7)).n(40) 1.983697455232980674869851942390639915940 We can rewrite harmonic numbers in terms of polygamma functions: >>> from sympy import digamma, polygamma >>> m = Symbol("m", integer=True, positive=True) >>> harmonic(n).rewrite(digamma) polygamma(0, n + 1) + EulerGamma >>> harmonic(n).rewrite(polygamma) polygamma(0, n + 1) + EulerGamma >>> harmonic(n,3).rewrite(polygamma) polygamma(2, n + 1)/2 + zeta(3) >>> simplify(harmonic(n,m).rewrite(polygamma)) Piecewise((polygamma(0, n + 1) + EulerGamma, Eq(m, 1)), (-(-1)**m*polygamma(m - 1, n + 1)/factorial(m - 1) + zeta(m), True)) Integer offsets in the argument can be pulled out: >>> from sympy import expand_func >>> expand_func(harmonic(n+4)) harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1) >>> expand_func(harmonic(n-4)) harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n Some limits can be computed as well: >>> from sympy import limit, oo >>> limit(harmonic(n), n, oo) oo >>> limit(harmonic(n, 2), n, oo) pi**2/6 >>> limit(harmonic(n, 3), n, oo) zeta(3) For `m > 1`, `H_{n,m}` tends to `\zeta(m)` in the limit of infinite `n`: >>> m = Symbol("m", positive=True) >>> limit(harmonic(n, m+1), n, oo) zeta(m + 1) See Also ======== bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci References ========== .. [1] https://en.wikipedia.org/wiki/Harmonic_number .. [2] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber/ .. [3] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/ z(dict[Integer, Callable[[int], Rational]]harmonic_cacheNc ^SSKJn T[RLaU"U5$Tc[RmUR(a[R $TR(aU$U[R LaxTR(a[R$[T[R5(a[R $[T[R5(aU"T5$gTR(a8TR(a'[ST- US-5[ST- 5- ST- - $UR(aUR(aQTRSLdTRSLa3T[RLa[R$[R$UR (aTR(aQTUR";a#[%S/5U4Sj5nX@R"T'UR"T"['U55$[)U4Sj[+S['U5S-556$gg)NrrrDFc<>US[RUT-- -$)Nr{)rr)rVrrs rHfharmonic.eval..fs#'8aeeadl#::rJc3.># UH oT*-v M g7fruri)rrrs rHr harmonic.eval..sC.B!W.Bs)rrrrrrrrrrrrrrrComplexInfinityrr*r=rr rE)rrVrrr-s ` rHr harmonic.evals? :q6M 9A 9966M YYH !**_}}uu q!%%zz!q!%%Aw! \\a..ac1Q3')AaC.8QqSA A \\}}!,,%"71;K;Ku;T,-Jq((AAEEA<< 2 22(!-;.;01**1---a0Q88CeAs1vz.BCDD rJc 2SSKJnJn UR(a~UR(al[ U"SUS-5[ R-[US54[ RU-U"US- S5U"US- US-5- -U"U5- S45$gg)Nr)gamma polygammarDT) 'sympy.functions.special.gamma_functionsr4r5rrr(r EulerGammarr)rrVrrr4r5s rH_eval_rewrite_as_polygamma#harmonic._eval_rewrite_as_polygammasL <1aI]]A%1Q3):YqsAaC=P)PQ!H"$% %*>coo8=c1b8IJ8I1aeeTX.8IJhW[nM]]F<'^^9>q#r9JK9JAquufdh/9JKxX\~N^^F<'}}}'')FAI## !--AEJIVVc ASa!eai1a)< isinstancer)rr is_nonzero) rrVrlimitvarrrr5pgargs rHr#harmonic._eval_rewrite_as_tractablesv?E \\) $"h''::k:5 5!%%i >>Gd1cl*33Kd3K K rJc[SUR55(dgURSRU5n[UR5S:aURSO[R RU5n[ R"U5(aUS:a[R$[U5 US:Xa[ R"U5nO0[ R"U5[ R"X2S-5- nSSS5 [R"WU5$!,(df  N%=f)Nc38# UHoRv M g7frurrs rHr'harmonic._eval_evalf..rrrrD)rrrrrrr?rrr@r)rr r)rrrVrrs rHrharmonic._eval_evalfs2 222  IIaL # #D ) ^a/TYYq\QUU F Ft L 88A;;1q555L d^Avkk!nggaj2771c?2    d++ ^s AD11 D?cSSKJn [UR5S:XaURup4OURS-up4US:XaXB"US-US-5-$[e)Nrrrr rD)rrrrr)rargindexrrVrs rHfdiffharmonic.fdiffsX? tyy>Q 99DAq99t#DA q=tAaC1~% %$ $rJrirur rDN)rjrkrlrmrnr*r rrrrr8r@rDr%rr]rrrnrprirJrHr)r)snAH@BN<AEE@/0ee%'''*+9 "HL , %rJr)cF\rSrSrSr\S Sj5rS SjrS SjrSr Sr g) euleri aL Euler numbers / Euler polynomials / Euler function The Euler numbers are given by: .. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j} \frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k} .. math:: E_{2n+1} = 0 Euler numbers and Euler polynomials are related by .. math:: E_n = 2^n E_n\left(\frac{1}{2}\right). We compute symbolic Euler polynomials using Appell sequences, but numerical evaluation of the Euler polynomial is computed more efficiently (and more accurately) using the mpmath library. The Euler polynomials are special cases of the generalized Euler function, related to the Genocchi function as .. math:: \operatorname{E}(s, a) = -\frac{\operatorname{G}(s+1, a)}{s+1} with the limit of `\psi\left(\frac{a+1}{2}\right) - \psi\left(\frac{a}{2}\right)` being taken when `s = -1`. The (ordinary) Euler function interpolating the Euler numbers is then obtained as `\operatorname{E}(s) = 2^s \operatorname{E}\left(s, \frac{1}{2}\right)`. * ``euler(n)`` gives the nth Euler number `E_n`. * ``euler(s)`` gives the Euler function `\operatorname{E}(s)`. * ``euler(n, x)`` gives the nth Euler polynomial `E_n(x)`. * ``euler(s, a)`` gives the generalized Euler function `\operatorname{E}(s, a)`. Examples ======== >>> from sympy import euler, Symbol, S >>> [euler(n) for n in range(10)] [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0] >>> [2**n*euler(n,1) for n in range(10)] [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936] >>> n = Symbol("n") >>> euler(n + 2*n) euler(3*n) >>> x = Symbol("x") >>> euler(n, x) euler(n, x) >>> euler(0, x) 1 >>> euler(1, x) x - 1/2 >>> euler(2, x) x**2 - x >>> euler(3, x) x**3 - 3*x**2/2 + 1/4 >>> euler(4, x) x**4 - 2*x**3 + x >>> euler(12, S.Half) 2702765/4096 >>> euler(12) 2702765 See Also ======== andre, bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi, partition, tribonacci, sympy.polys.appellseqs.euler_poly References ========== .. [1] https://en.wikipedia.org/wiki/Euler_numbers .. [2] https://mathworld.wolfram.com/EulerNumber.html .. [3] https://en.wikipedia.org/wiki/Alternating_permutation .. [4] https://mathworld.wolfram.com/AlternatingPermutation.html NcUR(a[R$U[RLa6Uc[RS- $SSKJn U"US-S- 5U"US- 5- $URSLdURSLagUcUR(a!UR(a[R$UR(a=SSK Jn URUR 5nUR""USS9n[%U5$gUR(aSS KJn [+U5nU"USS 9nU(a[-S U55(a[/S U55(a{SSK Jn [1UVs/sH!oR2(dMUR4PM# sn5n [7U 5 UR8"X5nSSS5 [:R<"WU 5$[?X5$gs snf!,(df  N6=f) NrrrJrDFr?T)exact) pure_complex)or_realc3^# UH#oR=(d URv M% g7fru)is_FloatrrrFs rHreuler.eval..xsE1JJ6!,,6s+-c38# UHoRv M g7fru)ryrzs rHrr{ys51JJr) rrrrrr6rKrrrrrrmpmathr?rreulernumrsympy.core.evalfrvrranyminry_precr@ eulerpolyr rr4) rrVrKrKr?rrvreimrFrs rHr euler.eval_sy 9955L !-- yttAv GAaC7#gacl2 2 \\U "a&6&6%&?  YxxAMMvv %LL)kk!40s|#  [[ 5AA40DEEEE5555%T@TZZGAGGT@Ad^,,q,C$((d33a# #A#^s;G3G35G88 Hc SSKJn UcUR(a[SSS9n[SSS9nUS- n[R U"U"[ XV5[RU-USU-- SU-S----SU-[R U--U-- USU45USSU-S-45-nU$U(aJ[SSS9nU"[ X5[U5-SU-- U[R- X- --USU45$g) Nrr rTrGrrrD) r$r!is_evenr rrr#rrrr)rrVrKrr!rrEms rHr%euler._eval_rewrite_as_Sums1 9c4(Ac4(AAA//CHQNammQ>N?@1Q3w!A#PQ'>R?S-Ta4**Q.-023Q)<>?AaC!G_%NNBI c4(Ax~eAh.q!t3QZ154IIAqRS9U U rJc FUcS[[RS- [US54SU-*[ US-[R 5-US-- S45$SSKJn [U"US-S- 5U"US- 5- [US54[ US-U5*US-- S45$)Nrr{rDTrrJ)r(rrrgenocchirr6rK)rrVrKrrKs rH_eval_rewrite_as_genocchieuler._eval_rewrite_as_genocchis 9add1fbBi0 !tehqsAFF&;;qsCTJL LC'1Q3'*WQqS\92a9E#AaC++qs3T:< Uc URO-UR"US-S- 5UR"US- 5- nOBUcSOUnSUR"U*U5SUS--UR"U*US-S- 5-- -nUcUSU--nSSS5 [R"WU5$!,(df  N%=f)Nc38# UHoRv M g7frurrrs rHr$euler._eval_evalf..rrrrtrDr{rr)rrr}r?rrr@rr~rrrKrr r)rrr?rrKrres rHreuler._eval_evalfsT2 222 '*499~': ! d#  LL  = T"A d^xx{{qAv() bkk!nr||A7I7#$9"%%"**ac1W2E STUVSV2WC yaArwwr1~AaC277A2!Qw;O0OOPC91a4KC  d++^s CF Friru) rjrkrlrmrnrrr%rrrprirJrHrrrr s.Ob$$B V<,rJrrcl\rSrSrSr\S5rSSjrSrSr SSjr Sr S r S r S rS rS rSrg)catalania} Catalan numbers The `n^{th}` catalan number is given by: .. math :: C_n = \frac{1}{n+1} \binom{2n}{n} * ``catalan(n)`` gives the `n^{th}` Catalan number, `C_n` Examples ======== >>> from sympy import (Symbol, binomial, gamma, hyper, ... catalan, diff, combsimp, Rational, I) >>> [catalan(i) for i in range(1,10)] [1, 2, 5, 14, 42, 132, 429, 1430, 4862] >>> n = Symbol("n", integer=True) >>> catalan(n) catalan(n) Catalan numbers can be transformed into several other, identical expressions involving other mathematical functions >>> catalan(n).rewrite(binomial) binomial(2*n, n)/(n + 1) >>> catalan(n).rewrite(gamma) 4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2)) >>> catalan(n).rewrite(hyper) hyper((-n, 1 - n), (2,), 1) For some non-integer values of n we can get closed form expressions by rewriting in terms of gamma functions: >>> catalan(Rational(1, 2)).rewrite(gamma) 8/(3*pi) We can differentiate the Catalan numbers C(n) interpreted as a continuous real function in n: >>> diff(catalan(n), n) (polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n) As a more advanced example consider the following ratio between consecutive numbers: >>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial)) 2*(2*n + 1)/(n + 2) The Catalan numbers can be generalized to complex numbers: >>> catalan(I).rewrite(gamma) 4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I)) and evaluated with arbitrary precision: >>> catalan(I).evalf(20) 0.39764993382373624267 - 0.020884341620842555705*I See Also ======== andre, bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci, sympy.functions.combinatorial.factorials.binomial References ========== .. [1] https://en.wikipedia.org/wiki/Catalan_number .. [2] https://mathworld.wolfram.com/CatalanNumber.html .. [3] https://functions.wolfram.com/GammaBetaErf/CatalanNumber/ .. [4] http://geometer.org/mathcircles/catalan.pdf cSSKJn UR(aUR(d"UR(aSUR (aBSU-U"U[ R-5-U"[ R5U"US-5-- $UR(aWUR (aEUS-R (a[ R$US-R(a [SS5$ggg)Nrr4rrrDr{) r6r4rr is_nonintegerrrrrrrr)rrVr4s rHr catalan.evalsA LLQ-- OO a4a!&&j))5=q1u+EF F LLQ]]A""vv AA&+LrJcSSKJn SSKJn URSn[ U5U"SU[ R-5U"SUS-5- U"S5--$)Nrr&r<rr)rSr'r6r5rrrr)rrmr'r5rVs rHrn catalan.fdiffsM>E IIaLqz9QAFF 3i1q56IICPQFRSSrJc ,[SU-U5US-- $NrrD)r#rs rH_eval_rewrite_as_binomial!catalan._eval_rewrite_as_binomials!QQ''rJc T[SU-5[US-5[U5-- $r)r$rs rH_eval_rewrite_as_factorial"catalan._eval_rewrite_as_factorials&1~1Q3)A,!>??rJc SSKJn SU-U"U[R-5-U"[R5U"US-5-- $)Nrrrr)r6r4rr)rrV piecewiserr4s rH_eval_rewrite_as_gammacatalan._eval_rewrite_as_gammas;A!tE!aff*%%uQVV}U1q5\'ABBrJc 0SSKJn U"SU- U*/S/S5$)Nr)hyperrDr)sympy.functions.special.hyperr)rrVrrs rH_eval_rewrite_as_hypercatalan._eval_rewrite_as_hyper"s 7a!eaR[1#q))rJc SSKJn UR(aUR(dU$[ SSSS9nU"X-U- USU45$)Nr)ProductrT)r"positiver)sympy.concrete.productsrrrr )rrVrrrs rH_eval_rewrite_as_Product catalan._eval_rewrite_as_Product&s@3 !1!1K #td 3{Q1I..rJcURSR(a URSR(agggNrTrrrrs rH_eval_is_integercatalan._eval_is_integer-s/ 99Q< " "tyy|'B'B(C "rJcBURSR(aggrrrrs rH_eval_is_positivecatalan._eval_is_positive1 99Q< & & 'rJcURSR(a#URSS- R(aggg)NrrT)rrrrs rH_eval_is_compositecatalan._eval_is_composite5s4 99Q< " " ! q(8'E'E(F "rJcSSKJn URSR(a UR U5R U5$g)Nrr)r6r4rrr>r)rrr4s rHrcatalan._eval_evalf9s5A 99Q< ! !<<&2248 8 "rJriNr )T)rjrkrlrmrnrrrnrrrrrrrrrrprirJrHrrsRM^ ' 'T (@C */9rJrcj\rSrSrSr\SSj5rSSjrSSjrSr Sr S r S r S r S rS rSrg)riFa Genocchi numbers / Genocchi polynomials / Genocchi function The Genocchi numbers are a sequence of integers `G_n` that satisfy the relation: .. math:: \frac{-2t}{1 + e^{-t}} = \sum_{n=0}^\infty \frac{G_n t^n}{n!} They are related to the Bernoulli numbers by .. math:: G_n = 2 (1 - 2^n) B_n and generalize like the Bernoulli numbers to the Genocchi polynomials and function as .. math:: \operatorname{G}(s, a) = 2 \left(\operatorname{B}(s, a) - 2^s \operatorname{B}\left(s, \frac{a+1}{2}\right)\right) .. versionchanged:: 1.12 ``genocchi(1)`` gives `-1` instead of `1`. Examples ======== >>> from sympy import genocchi, Symbol >>> [genocchi(n) for n in range(9)] [0, -1, -1, 0, 1, 0, -3, 0, 17] >>> n = Symbol('n', integer=True, positive=True) >>> genocchi(2*n + 1) 0 >>> x = Symbol('x') >>> genocchi(4, x) -4*x**3 + 6*x**2 - 1 See Also ======== bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci sympy.polys.appellseqs.genocchi_poly References ========== .. [1] https://en.wikipedia.org/wiki/Genocchi_number .. [2] https://mathworld.wolfram.com/GenocchiNumber.html .. [3] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 NcU[RLaU"U5$URSLdURSLagUcgUR(a$US- R (a[R $UR(a SS[S5U-- -[U5-$gUR(a [X5$g)NFrDr) rrrrrrrrrr5)rrVrKs rHr genocchi.evalys :q6M \\U "a&6&6%&?  YxxQqS--vv AadAgI155[[ & &rJc US:XaBUR(a1UR(a SS[S5U-- -[U5-$S[X5SU-[XS-S- 5-- -$r)rrrr)rrVrKrs rH_eval_rewrite_as_bernoulli#genocchi._eval_rewrite_as_bernoullis` 6allq'7'7!A$' ?Yq\1 1IaOadYqQ3!)-D&DDEErJc 2SSKJn SU-U"SU- U5-$)Nr) dirichlet_etarzrD)rr)rrVrKrrs rH_eval_rewrite_as_dirichlet_eta'genocchi._eval_rewrite_as_dirichlet_etasH!tmAaC+++rJc[UR5S:aURSS:wagURSnUR(aUR(aggg)NrDrT)rrrrrrVs rHrgenocchi._eval_is_integersJ tyy>A $))A,!"3  IIaL <A $))A,!"3  IIaL <A $))A,!"3  IIaL <UR(a,UR(a UR $US- R $ggNrDr)rrrrrrrrs rH _eval_is_evengenocchi._eval_is_evensh tyy>A $))A,!"3  IIaL <[UR5S:aURSS:wagURSnUR(aPUR(a>UR(a[ UR 5$[ US- R5$ggr)rrrrrrrrrs rH _eval_is_oddgenocchi._eval_is_oddsr tyy>A $))A,!"3  IIaL <A $))A,!"3  IIaL!}}rJc[SUR55(a$UR[5R U5$g)Nc38# UHoRv M g7frurrs rHr'genocchi._eval_evalf..s.Iq{{Ir)rrr>rr)rrs rHrgenocchi._eval_evalfs6 .DII. . .<< *66t< < /rJrirur )rjrkrlrmrnrrrrrrrrrrrrprirJrHrrFsJ0d ' 'F , !%0=rJrcF\rSrSrSr\S5rSrSrSr Sr Sr S r g ) andreia Andre numbers / Andre function The Andre number `\mathcal{A}_n` is Luschny's name for half the number of *alternating permutations* on `n` elements, where a permutation is alternating if adjacent elements alternately compare "greater" and "smaller" going from left to right. For example, `2 < 3 > 1 < 4` is an alternating permutation. This sequence is A000111 in the OEIS, which assigns the names *up/down numbers* and *Euler zigzag numbers*. It satisfies a recurrence relation similar to that for the Catalan numbers, with `\mathcal{A}_0 = 1` and .. math:: 2 \mathcal{A}_{n+1} = \sum_{k=0}^n \binom{n}{k} \mathcal{A}_k \mathcal{A}_{n-k} The Bernoulli and Euler numbers are signed transformations of the odd- and even-indexed elements of this sequence respectively: .. math :: \operatorname{B}_{2k} = \frac{2k \mathcal{A}_{2k-1}}{(-4)^k - (-16)^k} .. math :: \operatorname{E}_{2k} = (-1)^k \mathcal{A}_{2k} Like the Bernoulli and Euler numbers, the Andre numbers are interpolated by the entire Andre function: .. math :: \mathcal{A}(s) = (-i)^{s+1} \operatorname{Li}_{-s}(i) + i^{s+1} \operatorname{Li}_{-s}(-i) = \\ \frac{2 \Gamma(s+1)}{(2\pi)^{s+1}} (\zeta(s+1, 1/4) - \zeta(s+1, 3/4) \cos{\pi s}) Examples ======== >>> from sympy import andre, euler, bernoulli >>> [andre(n) for n in range(11)] [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521] >>> [(-1)**k * andre(2*k) for k in range(7)] [1, -1, 5, -61, 1385, -50521, 2702765] >>> [euler(2*k) for k in range(7)] [1, -1, 5, -61, 1385, -50521, 2702765] >>> [andre(2*k-1) * (2*k) / ((-4)**k - (-16)**k) for k in range(1, 8)] [1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6] >>> [bernoulli(2*k) for k in range(1, 8)] [1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6] See Also ======== bernoulli, catalan, euler, sympy.polys.appellseqs.andre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Alternating_permutation .. [2] https://mathworld.wolfram.com/EulerZigzagNumber.html .. [3] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 cDU[RLa[R$U[RLa[R$UR(a[R$US:Xa [ S5*$US:XaS[R -$UR(a}UR(a%UR(a[[U55$UR(a5SSK Jn U*S- n[U-[!SSU-- SU-5-U"U*5-$gg)Nr{rrzrrrDr)rrrrrr'Catalanrrrabsrrrrrrr)rrVrrs rHr andre.eval s :55L !**_::  9955L "WF7N "Waii<  \\AII58}$GBqD!thqAvq!t44tQBx??rJc SSKJn SSKJn SSKJn SU"US-5-S[ -US--- U"US-[RS- 5U"[ U-5U"US-[S5S- 5-- -$) Nr)rrrrrDrr) rTrr6r4rrrrr)rrrrr4rs rHrandre._eval_rewrite_as_zeta sr@A?51:~21 -ac1557#c"Q$i$qsAaDF2C&CCE ErJc SSKJn [*US--U"U*[5-[US--U"U*[*5--$)Nr)polylogrD)rrr)rrrrs rH_eval_rewrite_as_polylogandre._eval_rewrite_as_polylog's>Bac{WaR^+a!A#h!aR.HHHrJcjURSnUR(aUR(agggrrrs rHrandre._eval_is_integer+s( IIaL <>> from sympy import partition, Symbol >>> [partition(n) for n in range(9)] [1, 1, 2, 3, 5, 7, 11, 15, 22] >>> n = Symbol('n', integer=True, negative=True) >>> partition(n) 0 See Also ======== bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci References ========== .. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29 .. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem Tc2URSLa [S5eURSLa[R$UR SLdU[R La[R $URSLa[[[U555$g)NFn should be an integerT) r TypeErrorrrrrrrr/r>rs rHrpartition.evalgss <<5 45 5 ==D 66M 99 QUU 55L <<4 Zq *+ + rJc>URSRSLaggrrrs rHrpartition._eval_is_positivers 99Q< & &$ . /rJczURSnSH&up4X:XdM X!-U:XdM[Rs $ g)Nr))rr)r) r)rrr)rrrVrZrems rH _eval_Modpartition._eval_Modvs1 IIaL/FAv!%3,vv 0rJriN) rjrkrlrmrnrrrrrrrprirJrHrrDs/>JN,,rJrcJ\rSrSrSrSrSr\\R4Sj5r Sr g) divisor_sigmai~a Calculate the divisor function `\sigma_k(n)` for positive integer n ``divisor_sigma(n, k)`` is equal to ``sum([x**k for x in divisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k(n) = \prod_{i=1}^\omega (1+p_i^k+p_i^{2k}+\cdots + p_i^{m_ik}). Examples ======== >>> from sympy.functions.combinatorial.numbers import divisor_sigma >>> divisor_sigma(18, 0) 6 >>> divisor_sigma(39, 1) 56 >>> divisor_sigma(12, 2) 210 >>> divisor_sigma(37) 38 See Also ======== sympy.ntheory.factor_.divisor_count, totient, sympy.ntheory.factor_.divisors, sympy.ntheory.factor_.factorint References ========== .. [1] https://en.wikipedia.org/wiki/Divisor_function Tc (URSLa [S5eURSLa [S5eURSLa [S5eURSLa [S5eUR SLaSX--$U[ RLa[ R$URSLaURSLa2[[U5R5Vs/sHo3S-PM sn6$URSLa'[ [[U5[U555$URSLaP[[U5R5VVs/sH#upC[!XBUS---S- XB-S- - 5PM% snn6$ggs snfs snnfNFrn should be a positive integerzk should be an integerz!k should be a nonnegative integerTrD)rrrrrr^rrrrrr)valuesr*r>itemsr6)rrVrerZs rHrdivisor_sigma.evalsd <<5 45 5 ==E !=> > <<5 45 5  u $@A A :: qt8O :55L <<4 yyD IaL,?,?,AB,AqU,ABCC||t#q 6!9=>>yyE!S\]^S_SeSeSghSg41VQAE^a%7AD1H$EFSghii" Bis F *F riN rjrkrlrmrnrrrrrrrprirJrHr r ~s/'PJKuujjrJr cJ\rSrSrSrSrSr\\R4Sj5r Sr g)udivisor_sigmaia Calculate the unitary divisor function `\sigma_k^*(n)` for positive integer n ``udivisor_sigma(n, k)`` is equal to ``sum([x**k for x in udivisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k^*(n) = \prod_{i=1}^\omega (1+ p_i^{m_ik}). Parameters ========== k : power of divisors in the sum for k = 0, 1: ``udivisor_sigma(n, 0)`` is equal to ``udivisor_count(n)`` ``udivisor_sigma(n, 1)`` is equal to ``sum(udivisors(n))`` Default for k is 1. Examples ======== >>> from sympy.functions.combinatorial.numbers import udivisor_sigma >>> udivisor_sigma(18, 0) 4 >>> udivisor_sigma(74, 1) 114 >>> udivisor_sigma(36, 3) 47450 >>> udivisor_sigma(111) 152 See Also ======== sympy.ntheory.factor_.divisor_count, totient, sympy.ntheory.factor_.divisors, sympy.ntheory.factor_.udivisors, sympy.ntheory.factor_.udivisor_count, divisor_sigma, sympy.ntheory.factor_.factorint References ========== .. [1] https://mathworld.wolfram.com/UnitaryDivisorFunction.html Tc URSLa [S5eURSLa [S5eURSLa [S5eURSLa [S5eUR SLaSX--$UR (a<[[U5R5VVs/sHup4SX2U---PM snn6$gs snnfr ) rrrrrr^rrr)r)rrVrrZrs rHrudivisor_sigma.evals <<5 45 5 ==E !=> > <<5 45 5  u $@A A :: qt8O << ! 0B0B0DE0D1s80DEF F Es1C riNrrirJrHrrs/3hJKuu G GrJrc0\rSrSrSrSrSr\S5rSr g)legendre_symboli a Returns the Legendre symbol `(a / p)`. For an integer ``a`` and an odd prime ``p``, the Legendre symbol is defined as .. math :: \genfrac(){}{}{a}{p} = \begin{cases} 0 & \text{if } p \text{ divides } a\\ 1 & \text{if } a \text{ is a quadratic residue modulo } p\\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p \end{cases} Examples ======== >>> from sympy.functions.combinatorial.numbers import legendre_symbol >>> [legendre_symbol(i, 7) for i in range(7)] [0, 1, 1, -1, 1, -1, -1] >>> sorted(set([i**2 % 7 for i in range(7)])) [0, 1, 2, 4] See Also ======== sympy.ntheory.residue_ntheory.is_quad_residue, jacobi_symbol TFcURSLa [S5eURSLa [S5eURSLdURSLa [ S5eX-R SLa[ R$U[ RLa[ R$URSLa7URSLa'[ [[U5[U555$gg)NFa should be an integerzp should be an integerz p should be an odd prime integerT) rrr^rrrrrrrr r>)rrFrZs rHrlegendre_symbol.eval)s <<5 45 5 <<5 45 5 :: !((e"3?@ @ E??d "66M :55L <<4 ALLD$8XfQi34 4%9 rJriN rjrkrlrmrnrr^rrrprirJrHrr s%8JH 5 5rJrc0\rSrSrSrSrSr\S5rSr g) jacobi_symboli9a Returns the Jacobi symbol `(m / n)`. For any integer ``m`` and any positive odd integer ``n`` the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of ``n``: .. math :: \genfrac(){}{}{m}{n} = \genfrac(){}{}{m}{p^{1}}^{\alpha_1} \genfrac(){}{}{m}{p^{2}}^{\alpha_2} ... \genfrac(){}{}{m}{p^{k}}^{\alpha_k} \text{ where } n = p_1^{\alpha_1} p_2^{\alpha_2} ... p_k^{\alpha_k} Like the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = -1` then ``m`` is a quadratic nonresidue modulo ``n``. But, unlike the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = 1` then ``m`` may or may not be a quadratic residue modulo ``n``. Examples ======== >>> from sympy.functions.combinatorial.numbers import jacobi_symbol, legendre_symbol >>> from sympy import S >>> jacobi_symbol(45, 77) -1 >>> jacobi_symbol(60, 121) 1 The relationship between the ``jacobi_symbol`` and ``legendre_symbol`` can be demonstrated as follows: >>> L = legendre_symbol >>> S(45).factors() {3: 2, 5: 1} >>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1 True See Also ======== sympy.ntheory.residue_ntheory.is_quad_residue, legendre_symbol TFcURSLa [S5eURSLa [S5eURSLdURSLa [ S5eU[ R LdU[ R La[ R $X-RSLa[ R$URSLa7URSLa'[ [[U5[U555$gg)NFzm should be an integerrz#n should be an odd positive integerT) rrrrrrrrrrr!r>)rrrVs rHrjacobi_symbol.evalps <<5 45 5 <<5 45 5 ==E !QXX%6BC C :aee55L E??d "66M <<4 ALLD$8VF1Ivay12 2%9 rJriNrrirJrHrr9s&2fJH 3 3rJrc0\rSrSrSrSrSr\S5rSr g)kronecker_symboliaG Returns the Kronecker symbol `(a / n)`. Examples ======== >>> from sympy.functions.combinatorial.numbers import kronecker_symbol >>> kronecker_symbol(45, 77) -1 >>> kronecker_symbol(13, -120) 1 See Also ======== jacobi_symbol, legendre_symbol References ========== .. [1] https://en.wikipedia.org/wiki/Kronecker_symbol TFcdURSLa [S5eURSLa [S5eU[RLdU[RLa[R$URSLa7URSLa'[[ [ U5[ U555$gg)NFrrT)rrrrrr"r>)rrFrVs rHrkronecker_symbol.evals <<5 45 5 <<5 45 5 :aee55L <<4 ALLD$8Yvay&)45 5%9 rJriNrrirJrHr!r!s%.JH66rJr!c0\rSrSrSrSrSr\S5rSr g)mobiusia  Mobius function maps natural number to {-1, 0, 1} It is defined as follows: 1) `1` if `n = 1`. 2) `0` if `n` has a squared prime factor. 3) `(-1)^k` if `n` is a square-free positive integer with `k` number of prime factors. It is an important multiplicative function in number theory and combinatorics. It has applications in mathematical series, algebraic number theory and also physics (Fermion operator has very concrete realization with Mobius Function model). Examples ======== >>> from sympy.functions.combinatorial.numbers import mobius >>> mobius(13*7) 1 >>> mobius(1) 1 >>> mobius(13*7*5) -1 >>> mobius(13**2) 0 Even in the case of a symbol, if it clearly contains a squared prime factor, it will be zero. >>> from sympy import Symbol >>> n = Symbol("n", integer=True, positive=True) >>> mobius(4*n) 0 >>> mobius(n**2) 0 References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_function .. [2] Thomas Koshy "Elementary Number Theory with Applications" .. [3] https://oeis.org/A008683 TFcZURSLa [S5eURSLa [S5eURSLa[ R $U[ RLa[ R$SnS[R"U55GHup4URSLaURSLaURSLaURSLa[[ RU5nUSLa[ RnMsURSLa[U5n[SUR555(a[ RnMUSLaB[!U5S-(a[ R O[ RnUcUnGMX'-nGM GMGM g U$)NFrr Tc3@# UHoR5v M g7fru) as_base_exp)r_s rHrmobius.eval..s?.>]]__.>c3*# UH oS:v M g7frprirvs rHrr*s;*:Qq5*:r)rrrrr^rrrr make_argsrrrr)rr r)rrVrXrrltfactorsrs rHr mobius.evalsG <<5 45 5 ==E !=> > :: == :55L?cmmA.>?DA||t# (=||t# (=155!_:VVF\\T)'lG;'..*:;;;!"u-0\A-=AMM155!>%&F"KF % *#@$ rJriNrrirJrHr%r%s&+XJHrJr%c0\rSrSrSrSrSr\S5rSr g)primenuia Calculate the number of distinct prime factors for a positive integer n. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primenu(n)`` or `\nu(n)` is: .. math :: \nu(n) = k. Examples ======== >>> from sympy.functions.combinatorial.numbers import primenu >>> primenu(1) 0 >>> primenu(30) 3 See Also ======== sympy.ntheory.factor_.factorint References ========== .. [1] https://mathworld.wolfram.com/PrimeFactor.html .. [2] https://oeis.org/A001221 TcHURSLa [S5eURSLa [S5eURSLa[ R $U[ R La[ R$URSLa[ [[U555$gNFrr T) rrrrr^rrrrrr)rs rHr primenu.evals <<5 45 5 ==E !=> > :: 55L :66M <<4 S1&' ' rJriN rjrkrlrmrnrrrrrprirJrHr5r5s&!DJN ( (rJr5c0\rSrSrSrSrSr\S5rSr g) primeomegai+a! Calculate the number of prime factors counting multiplicities for a positive integer n. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primeomega(n)`` or `\Omega(n)` is: .. math :: \Omega(n) = \sum_{i=1}^k m_i. Examples ======== >>> from sympy.functions.combinatorial.numbers import primeomega >>> primeomega(1) 0 >>> primeomega(20) 3 See Also ======== sympy.ntheory.factor_.factorint References ========== .. [1] https://mathworld.wolfram.com/PrimeFactor.html .. [2] https://oeis.org/A001222 TcdURSLa [S5eURSLa [S5eURSLa[ R $U[ R La[ R$URSLa+[ [[U5R555$gr7) rrrrr^rrrrsumr)r rs rHrprimeomega.evalRs <<5 45 5 ==E !=> > :: 55L :66M <<4 S1,,./0 0 rJriNr9rirJrHr;r;+s&"FJN 1 1rJr;c0\rSrSrSrSrSr\S5rSr g)totienti`a% Calculate the Euler totient function phi(n) ``totient(n)`` or `\phi(n)` is the number of positive integers `\leq` n that are relatively prime to n. Examples ======== >>> from sympy.functions.combinatorial.numbers import totient >>> totient(1) 1 >>> totient(25) 20 >>> totient(45) == totient(5)*totient(9) True See Also ======== sympy.ntheory.factor_.divisor_count References ========== .. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function .. [2] https://mathworld.wolfram.com/TotientFunction.html .. [3] https://oeis.org/A000010 TcURSLa [S5eURSLa [S5eU[R La[R $UR SLaUS- $[U[5(a)[ [SUR5555$URSLa2[ [S[U5R5555$g)NFrr TrDc3@# UHupXS- -US- -v M g7frprirrZrs rHrtotient.eval..s ?YTQ!c(AaC.Yr+c3@# UHupXS- -US- -v M g7frprirCs rHrrDs"J5ITQ!c(AaC.5Ir+) rrrrrrr^rbr rrrr)rs rHr totient.evals <<5 45 5 ==E !=> > :55L :: q5L a  T?QWWY??@ @ <<4 TJYq\5G5G5IJJK K rJriN rjrkrlrmrnrrrrrprirJrHr@r@`s'<JK L LrJr@c0\rSrSrSrSrSr\S5rSr g)reduced_totientia- Calculate the Carmichael reduced totient function lambda(n) ``reduced_totient(n)`` or `\lambda(n)` is the smallest m > 0 such that `k^m \equiv 1 \mod n` for all k relatively prime to n. Examples ======== >>> from sympy.functions.combinatorial.numbers import reduced_totient >>> reduced_totient(1) 1 >>> reduced_totient(8) 2 >>> reduced_totient(30) 4 See Also ======== totient References ========== .. [1] https://en.wikipedia.org/wiki/Carmichael_function .. [2] https://mathworld.wolfram.com/CarmichaelFunction.html .. [3] https://oeis.org/A002322 Tc URSLa [S5eURSLa [S5eU[R La[R $UR SLaUS- $[U[5(aTSnSU;aSUS:a SUSS- -OUSn[ [[U5/SUR55Q765$URSLab[[U5S5upU(dSnOSU:aSUS- -n[ [U/S[U5R55Q765$g) NFrr TrDrc3# UH8upUS:wdM [US- 5[U5[US- 5--v M: g7f)rrDN)rrCs rHr'reduced_totient.eval..s<"]VW[\V\#<3qs8CFC!H,<#<s A0Ac3@# UHupUS- XS- --v M g7frprirCs rHrrLs"N9MqsA!Hn9Mr+)rrrrrrr^rbr rrrrrr))rrVts rHrreduced_totient.evals! <<5 45 5 ==E !=> > :55L :: q5L a  AAv)*QqTQ1Q4!8_qtSQ^"]"]^_ _ <<4 #a&!$DAQ!a%LSON19K9K9MNOP P rJriNrGrirJrHrIrIs'<JKQQrJrIc0\rSrSrSrSrSr\S5rSr g)primepiiaRepresents the prime counting function pi(n) = the number of prime numbers less than or equal to n. Examples ======== >>> from sympy.functions.combinatorial.numbers import primepi >>> from sympy import prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime sympy.ntheory.generate.primerange : Generate all primes in a given range sympy.ntheory.generate.prime : Return the nth prime References ========== .. [1] https://oeis.org/A000720 TcrU[RLa[R$U[RLa[R$URSLa [ S5e[ U[S55SLa[R$[U5n[[U55$![ a gf=f)NFzn should be a realrT) rrNegativeInfinityris_realrrrr.rs rHr primepi.evals  ?::  "" "66M 99 01 1 AaD>T !66M AA!~   s B)) B65B6riNr9rirJrHrQrQs&"FJN  rJrQc\rSrSrSrSrg)_MultisetHistogrami riN)rjrkrlrm __slots__rprirJrHrWrW sIrJrWr{rzNc^[T[5(a[STR555(d[e[ TR55n[ U4SjT55n[ TVs/sHnTUS:dMTUPM snX!/-5$[T5m[T5n[U5n[T5nXV:XaS/U-Xf/-m[ T5$[[U[U555n[[[U5SU-55nTHn XU ==S- ss'M [U5$s snf)a"Return tuple used in permutation and combination counting. Input is a dictionary giving items with counts as values or a sequence of items (which need not be sorted). The data is stored in a class deriving from tuple so it is easily recognized and so it can be converted easily to a list. c3Z# UH!n[U[5=(a US:v M# g7f)rN)rbrr-s rHr&_multiset_histogram.. s#E*Q:a%0!q&0*s)+c3>># UHnTUS:dMSv M g7f)rrDNri)rrrVs rHrr[ s-q!AaD1HAAqs  rrDr) rbdictrr rr=rWlistsetrziprE_multiset_histogram) rVtotrrrlenslennrdrs ` rHrbrb s!TE!((*EEE !((*o-q--!"?AadQh41Q4"?5,"NOO G F1v1v <DD<'A%a( ( Qd $ % U4[$t), -A dGqLG"1%%#@s 9 E Ec[U5n[[XU55$![a! [[[ U5X55s$f=f)aReturn the number of permutations of ``n`` items taken ``k`` at a time. Possible values for ``n``: integer - set of length ``n`` sequence - converted to a multiset internally multiset - {element: multiplicity} If ``k`` is None then the total of all permutations of length 0 through the number of items represented by ``n`` will be returned. If ``replacement`` is True then a given item can appear more than once in the ``k`` items. (For example, for 'ab' permutations of 2 would include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in ``n`` is ignored when ``replacement`` is True but the total number of elements is considered since no element can appear more times than the number of elements in ``n``. Examples ======== >>> from sympy.functions.combinatorial.numbers import nP >>> from sympy.utilities.iterables import multiset_permutations, multiset >>> nP(3, 2) 6 >>> nP('abc', 2) == nP(multiset('abc'), 2) == 6 True >>> nP('aab', 2) 3 >>> nP([1, 2, 2], 2) 3 >>> [nP(3, i) for i in range(4)] [1, 3, 6, 6] >>> nP(3) == sum(_) True When ``replacement`` is True, each item can have multiplicity equal to the length represented by ``n``: >>> nP('aabc', replacement=True) 121 >>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)] [1, 3, 9, 27, 81] >>> sum(_) 121 See Also ======== sympy.utilities.iterables.multiset_permutations References ========== .. [1] https://en.wikipedia.org/wiki/Permutation )r>rr_nPrb)rVr replacements rHnPrj1 sQvD 1I 3q[) ** Ds.q11BCCDs "(A  A c V^^US:Xag[T[5(acUc"[UU4Sj[TS-555$T(aTU-$UT:agUT:Xa [ U5$US:XaT$[ TU- S-T5$[T[ 5(GaUc)[UU4Sj[T[S-555$T(a T[U-$UT[:Xa?[ U5[T[Vs/sHo3S:dM [ U5PM sn5- $UT[:agUS:Xa T[$Sn[T5m[[T[55HnTU(dMT[==S-ss'TUS:XaHSTU'T[==S-ss'U[[ T5US- 5- nT[==S- ss'STU'O5TU==S-ss'U[[ T5US- 5- nTU==S- ss'T[==S- ss'M U$gs snf)NrrDc3>># UHn[TUT5v M g7frurhrrrVris rHr_nP..{ sD|!s1a--|c3>># UHn[TUT5v M g7frurmrns rHrro sH7G!s1a--7Grp)rbrr=rEr$rIrW_N_ITEMSr_Mr_rrh)rVrrirrcs` ` rHrhrhs s Av!Z  9DuQU|DD D a4K U !VQ<  !VHAEAIq) ) A) * * 9HuQrUQY7GHH H V9a<  !B%ZQ<AbE%KEqUlilE%K LL L 2Y !VV9 CQA3qu:&t" Q419AaDfINI31!4a!e< H& H& c n[U5n[U5nUS-S-nS/UR5S--nURSU[ U5- -5 USUnU(aUR5nUS-nUSSn[ S[ U[ U555HnX7==X7S- - ss'M [ XR5HnX7==X7S- XgU- - - ss'M U(aM[[U55nUS-(aX8-nOXSS&[[5n [U5H upzXU'M U $)afor n = (m1, m2, .., mk) return the coefficients of the polynomial, prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients of the product of AOPs (all-one polynomials) or order given in n. The resulting coefficient corresponding to x**r is the number of r-length combinations of sum(n) elements with multiplicities given in n. The coefficients are given as a default dictionary (so if a query is made for a key that is not present, 0 will be returned). Examples ======== >>> from sympy.functions.combinatorial.numbers import _AOP_product >>> from sympy.abc import x >>> n = (2, 2, 3) # e.g. aabbccc >>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand() >>> c = _AOP_product(n); dict(c) {0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1} >>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)] True The generating poly used here is the same as that listed in https://tinyurl.com/cep849r, but in a refactored form. rrDr]Nr{) r_r=popextendrrErreversedrr enumerate) rVordneedrvniNwasrrevrfrs rH _AOP_productr s)6 QA a&C !Ga>> from sympy.functions.combinatorial.numbers import nC >>> from sympy.utilities.iterables import multiset_combinations >>> nC(3, 2) 3 >>> nC('abc', 2) 3 >>> nC('aab', 2) 2 When ``replacement`` is True, each item can have multiplicity equal to the length represented by ``n``: >>> nC('aabc', replacement=True) 35 >>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)] [1, 3, 6, 10, 15] >>> sum(_) 35 If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k`` then the total of all combinations of length 0 through ``k`` is the product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity of each item is 1 (i.e., k unique items) then there are 2**k combinations. For example, if there are 4 unique items, the total number of combinations is 16: >>> sum(nC(4, i) for i in range(5)) 16 See Also ======== sympy.utilities.iterables.multiset_combinations References ========== .. [1] https://en.wikipedia.org/wiki/Combination .. [2] https://tinyurl.com/cep849r rc3>># UHn[TUT5v M g7frunCrns rHrnC.." Clr!Q ,,lrprDrzk cannot be negativec3*# UH oS-v M g7frpri)rrs rHrr, s15aE5r/c3>># UHn[TUT5v M g7frurrns rHrr- rrp)rbrr=rErr#rWrrrrtrrsrtuplerb)rVrrir~s` ` rHrr s+D!Z  9!t CeAElCC C q534 4 AEAIq) )1~!'(( bE 911R5111CeAElCC C aiK0 0 1a!e*_V9  1a&[E!B%L)!,,%a(![99rJcVXs=:XaS:Xa[R$ SX4;a[R$X:Xa[R$XS- :Xa [US5$XS- :XaSU-S- [US5-S- $XS- :Xa[US5[US5-$[ X5$)NrrDrrr)rrrr# _stirling1rVrs rH_eval_stirling1r: s{{uu QF{vv  vuu !e1~ !e!a!Q')) !e1~hq!n,, a rJcSS/S/US- --n[SUS-5H4n[[X5SS5HnUS- X$-X$S- -X$'M M6 [X!5$NrrDrr{rErrrVrrowrrs rHrrM so a&!ac C 1ac]s1xB'AcSV^cA#h.CF( 36?rJc4Xs=:XaS:Xa[R$ SX4;a[R$X:Xa[R$XS- :Xa [US5$US:Xa[R$US:Xa[ SUS- -S- 5$[ X5$)NrrDr)rrrr#r _stirling2rs rH_eval_stirling2rV s{{uu QF{vv  vuu !e1~ auu aq1q5zA~&& a rJcSS/S/US- --n[SUS-5H1n[[X5SS5HnXBU-X$S- -X$'M M3 [X!5$rrrs rHrri sk a&!ac C 1ac]s1xB'AQZ#c(*CF( 36?rJcd[U5n[U5nUS:a [S5eX:a[R$U(a[ X- S-X- S-5$U(a![R X- -[ X5-$US:Xa [ X5$US:Xa [ X5$[SU-5e)a Return Stirling number $S(n, k)$ of the first or second (default) kind. The sum of all Stirling numbers of the second kind for $k = 1$ through $n$ is ``bell(n)``. The recurrence relationship for these numbers is: .. math :: {0 \brace 0} = 1; {n \brace 0} = {0 \brace k} = 0; .. math :: {{n+1} \brace k} = j {n \brace k} + {n \brace {k-1}} where $j$ is: $n$ for Stirling numbers of the first kind, $-n$ for signed Stirling numbers of the first kind, $k$ for Stirling numbers of the second kind. The first kind of Stirling number counts the number of permutations of ``n`` distinct items that have ``k`` cycles; the second kind counts the ways in which ``n`` distinct items can be partitioned into ``k`` parts. If ``d`` is given, the "reduced Stirling number of the second kind" is returned: $S^{d}(n, k) = S(n - d + 1, k - d + 1)$ with $n \ge k \ge d$. (This counts the ways to partition $n$ consecutive integers into $k$ groups with no pairwise difference less than $d$. See example below.) To obtain the signed Stirling numbers of the first kind, use keyword ``signed=True``. Using this keyword automatically sets ``kind`` to 1. Examples ======== >>> from sympy.functions.combinatorial.numbers import stirling, bell >>> from sympy.combinatorics import Permutation >>> from sympy.utilities.iterables import multiset_partitions, permutations First kind (unsigned by default): >>> [stirling(6, i, kind=1) for i in range(7)] [0, 120, 274, 225, 85, 15, 1] >>> perms = list(permutations(range(4))) >>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)] [0, 6, 11, 6, 1] >>> [stirling(4, i, kind=1) for i in range(5)] [0, 6, 11, 6, 1] First kind (signed): >>> [stirling(4, i, signed=True) for i in range(5)] [0, -6, 11, -6, 1] Second kind: >>> [stirling(10, i) for i in range(12)] [0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0] >>> sum(_) == bell(10) True >>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2) True Reduced second kind: >>> from sympy import subsets, oo >>> def delta(p): ... if len(p) == 1: ... return oo ... return min(abs(i[0] - i[1]) for i in subsets(p, 2)) >>> parts = multiset_partitions(range(5), 3) >>> d = 2 >>> sum(1 for p in parts if all(delta(i) >= d for i in p)) 7 >>> stirling(5, 3, 2) 7 See Also ======== sympy.utilities.iterables.multiset_partitions References ========== .. [1] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind .. [2] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind rzn must be nonnegativerDrzkind must be 1 or 2, not %s)r>rrrrrr)rVrrfkindsigneds rHstirlingrr sl q Aq A1u011uvv quqy!%!)44 }}qu%oa&;;; qyq$$ q$$6:;;rJc X:dUS:agUSU4;agUS:XagUS:XaUS-$X- nUS::aU$SU-U:aB[U5nX!- S:a-U[[R"[X!- 555-nU$S/U-n[ SUS-5H/n[ US-U5HnXF==XFU- - ss'M US-nM1 S[USU- S5-$)ztReturn the partitions of ``n`` items into ``k`` parts. This is used by ``nT`` for the case when ``n`` is an integer.rrDrrN)r0r= fetch_itemslicerE)rVrrfrcrZrrs rH_nTr s  uAQF{AvAv!t  AAvsax Q 519 3~00qu>? ?C  AA 1a!e_Aq/A DA!eH D" Q Aa!efI rJc^[T[5(aNUc [T5$[U[5(a+[T5m[U5n[ [ TU55$[T[ 5(dP[[T55nUS::a[[T5U5$U[T5:Xa [U5m[eT[nUcUS:XagUSU4;agUS:Xd US:XaVUcS[US5upE[U4Sj[SUS-555nU(dU[!TU5S--nUcUS- nU$UT[":XaUc [%U5$['X15$[)5nUcUR+T[,5$SnUR/T[,US- U5HnUS- nM U$![a [T5mGNf=f)aReturn the number of ``k``-sized partitions of ``n`` items. Possible values for ``n``: integer - ``n`` identical items sequence - converted to a multiset internally multiset - {element: multiplicity} Note: the convention for ``nT`` is different than that of ``nC`` and ``nP`` in that here an integer indicates ``n`` *identical* items instead of a set of length ``n``; this is in keeping with the ``partitions`` function which treats its integer-``n`` input like a list of ``n`` 1s. One can use ``range(n)`` for ``n`` to indicate ``n`` distinct items. If ``k`` is None then the total number of ways to partition the elements represented in ``n`` will be returned. Examples ======== >>> from sympy.functions.combinatorial.numbers import nT Partitions of the given multiset: >>> [nT('aabbc', i) for i in range(1, 7)] [1, 8, 11, 5, 1, 0] >>> nT('aabbc') == sum(_) True >>> [nT("mississippi", i) for i in range(1, 12)] [1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1] Partitions when all items are identical: >>> [nT(5, i) for i in range(1, 6)] [1, 2, 2, 1, 1] >>> nT('1'*5) == sum(_) True When all items are different: >>> [nT(range(5), i) for i in range(1, 6)] [1, 15, 25, 10, 1] >>> nT(range(5)) == sum(_) True Partitions of an integer expressed as a sum of positive integers: >>> from sympy import partition >>> partition(4) 5 >>> nT(4, 1) + nT(4, 2) + nT(4, 3) + nT(4, 4) 5 >>> nT('1'*4) 5 See Also ======== sympy.utilities.iterables.partitions sympy.utilities.iterables.multiset_partitions sympy.functions.combinatorial.numbers.partition References ========== .. [1] https://web.archive.org/web/20210507012732/https://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf rDrc3<># UHn[TU5v M g7frur)rrrVs rHrnT..n s3?aAq?sr)rbrrr>rrrWrr`nTrErrbrrdivmodr=rrsr rr7count_partitionsrt enum_range) rVrrYr~rrr|rcdiscards ` rHrr sR!Z  9Q<  a $ $q Aq A3q!9% % a+ , , 'CF AAv#a&!}$c!f!HO "AyQ!VQF{AvaAIa| 35AE?3 3 "Q(A+ B 9 !GB AfI~ 97N~"$Ay!!!B%(( C<<"qsA. q/ J9 '#A&A 's;.G* GGGc\rSrSrSr\S5r\S5r\S5r\\ "\ R\ R/5S55r \ S5rSrg ) motzkini a The nth Motzkin number is the number of ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. Motzkin numbers are the integer sequence defined by the initial terms `M_0 = 1`, `M_1 = 1` and the two-term recurrence relation `M_n = rac{2*n + 1}{n + 2} * M_{n-1} + rac{3n - 3}{n + 2} * M_{n-2}`. Examples ======== >>> from sympy import motzkin >>> motzkin.is_motzkin(5) False >>> motzkin.find_motzkin_numbers_in_range(2,300) [2, 4, 9, 21, 51, 127] >>> motzkin.find_motzkin_numbers_in_range(2,900) [2, 4, 9, 21, 51, 127, 323, 835] >>> motzkin.find_first_n_motzkins(10) [1, 1, 2, 4, 9, 21, 51, 127, 323, 835] References ========== .. [1] https://en.wikipedia.org/wiki/Motzkin_number .. [2] https://mathworld.wolfram.com/MotzkinNumber.html c[U5nUS:aFUS;agSnSnSnX :a-SU-S-U-SU-S- U--US-- nUS- nUnUnX :aM-X :Xaggg![a gf=f)NFr)rDrTrDrr)r>r)rVtn1tnrrFs rH is_motzkinmotzkin.is_motzkin s q A q5F{CBA&cAgr\QqS1WcM1AE:Q & w+  s A A'&A'c0SUs=::aU::aO O|/nUSs=::aU::aO OURS5 SnSnSnXA::aLX@:aURU5 SU-S-U-SU-S- U--US-- nUS- nUn[U5nXA::aMLU$[S5e)NrrDrrzEThe provided range is not valid. This condition should satisfy x <= y)appendrr)rKremotzkinsrrrrFs rHfind_motzkin_numbers_in_range%motzkin.find_motzkin_numbers_in_range s ;Q;HA{{"CBA'7OOB'cAgr\QqS1WcM1AE:QV 'Ode erJcJ[U5nUS:a [S5eS/nUS:aURS5 SnSnSnX@::aGURU5 SU-S-U-SU-S- U--US-- nUS- nUn[U5nX@::aMGU$![a [S5ef=f)N.The provided number must be a positive integerrrDrr)r>rrr)rVrrrrrFs rHfind_first_n_motzkinsmotzkin.find_first_n_motzkins s Oq A q5MN N3 6 OOA   f OOB A#'21q# -A6A FACQB f# OMN N Os B B"cHSU-S-US-SU-S- US--US--$)NrrDr{rrzrirs rH_motzkinmotzkin._motzkin s:1q$r("acAgtBx%77QUCCrJc[U5nUS:a [S5e[URUS- 55$![a [S5ef=f)NrrrD)r>rrrrs rHr motzkin.eval s] Oq A q5MN Ns||AE*++  OMN N Os ;AriN)rjrkrlrmrnrorrrr=rrrrrrprirJrHrr s"H4ff*,aeeQUU^$D%D,,rJr)rVrch^SSKJn SSKJn SSKJn SmXU4R S5S:wa [S5eUGb[U[5(a [S 5eU(d[R$[U5[La[U5n[!U5nOd[U5[LaR[#U4S jUR%555 UR'5V V s0sHupU (dMX_M nn n SnW(d[R$[)UR%55n [!UR%55n [+U5n GO`Ub*T"U5 U(d[R$[-U5$UGb/[U[5(aS[#U4S jUR'555 UR'5V V s0sHupX-(dMX_M n n n Ou[/U5(d[U[05(aE[U5n[#U4S jU55 [!UVs/sH o(dM UPM sn5n O [S 5eU (d[R$[)SU R'555n [)U R%55n Sn[3[5W 55nUS:Xa [-W 5$[+U 5nUS-W :a[R$US-U :Xa0W S:XaUS:Xa[R6$U S- U:Xa [9U5$U S:aUbU(apWcI/nSnU R'5H1up:[;U 5HnUR=U/U-5 US- nM M3 [?[)S[AU5555$SSK!J"n [?[GU"U"U*5[IU R'5VVs/sHupU"X5U-PM snn6-US[J4555$s sn n fs sn n fs snfs snnf)ayreturn the number of derangements for: ``n`` unique items, ``i`` items (as a sequence or multiset), or multiplicities, ``m`` given as a sequence or multiset. Examples ======== >>> from sympy.utilities.iterables import generate_derangements as enum >>> from sympy.functions.combinatorial.numbers import nD A derangement ``d`` of sequence ``s`` has all ``d[i] != s[i]``: >>> set([''.join(i) for i in enum('abc')]) {'bca', 'cab'} >>> nD('abc') 2 Input as iterable or dictionary (multiset form) is accepted: >>> assert nD([1, 2, 2, 3, 3, 3]) == nD({1: 1, 2: 2, 3: 3}) By default, a brute-force enumeration and count of multiset permutations is only done if there are fewer than 9 elements. There may be cases when there is high multiplicity with few unique elements that will benefit from a brute-force enumeration, too. For this reason, the `brute` keyword (default None) is provided. When False, the brute-force enumeration will never be used. When True, it will always be used. >>> nD('1111222233', brute=True) 44 For convenience, one may specify ``n`` distinct items using the ``n`` keyword: >>> assert nD(n=3) == nD('abc') == 2 Since the number of derangments depends on the multiplicity of the elements and not the elements themselves, it may be more convenient to give a list or multiset of multiplicities using keyword ``m``: >>> assert nD('abc') == nD(m=(1,1,1)) == nD(m={1:3}) == 2 r) integrate)laguerrerKcf[U[5(d [S5eUS:a [S5eg)Nzexpecting integer valuesrzvalue must not be negativeT)rbrrrrs rHoknD..ok7 s1!Z((67 7 q59: :rJNrzenter only 1 of i, n, or mz"items must be a list or dictionaryc34># UH nT"U5v M g7fruri)rr)rs rHrnD..I s*z!1zc3V># UHupT"U5=(a T"U5v M g7fruri)rrrrs rHrrX s!6IDA1"Q%Is&)c34># UH nT"U5v M g7fruri)rrrs rHrr\ s!q!1qrzexpecting iterablec3.# UH upX-v M g7fruri)rrr.s rHrrb s/srDrc3&# UHnSv M g7frprirs rHrry s?&>1&>s)exp)&sympy.integrals.integralsr#sympy.functions.special.polynomialsr sympy.abcrKcountrrbrrrrtyper^r_r:rr rr=rr%r<strrmaxrr$rErwrr;rSrrrr)rbruterVrrrrKrmsrr.r~countsnkeybignvalrrrs @rHnDr sUX4< ayt!566} a $ $@A A66M 7$ QA!B !W_ *qxxz* *#$7792941$!$9B2A66M  "))+&2w  166MA  a   6AGGI6 6'(wwy8ytqACdadyF8F a[[Jq#..QA !q! !!1!Qqq!12F01 166M / / /6==?#  c&k C axD!! v;D 1uqyvv  1uz 1955L !8s?S> !1u% 9AA qAHHaSUOFA"'s?&;A&>??@@: 3yaR&,lln/6&4dan/6*7"79:Ar DE FFc392</6s*% P6PP#&P#> P) P)+P.)NF)NrFrur')rn __future__rmathr collectionsrtypingr sympy.corerrr r sympy.core.cacher sympy.core.containersr sympy.core.exprr sympy.core.functionrrrsympy.core.logicrsympy.core.mulrsympy.core.numbersrrrrrrsympy.core.relationalrrrrsympy.external.gmpyrrrr r!r"(sympy.functions.combinatorial.factorialsr#r$r%rSr'$sympy.functions.elementary.piecewiser(sympy.ntheory.factor_r)r*r+r,r-sympy.ntheory.generater.sympy.ntheory.partitions_r/r0sympy.ntheory.primetestr1r2sympy.polys.appellseqsr3r4r5sympy.polys.polytoolsr6sympy.utilities.enumerativer7sympy.utilities.exceptionsr9sympy.utilities.iterablesr:r;r<sympy.utilities.memoizationr=sympy.utilities.miscr>r}r?r@ mpmath.libmprArvrIr}rMrrrrrr r)rrrrrrr rrrr!r%r5r;r@rIrQrrWrrrsrrtrbrjrhrrrrrrrrrrrrirJrHrs##,,$& OO&>>99TT6:__+@6LL(B@OO7'&! c{x+x+DNNNNp*>O*>hP"P"tt,t,|OD?ODrI%I%f],O],NH9oH9dB=B=Xi,Oi,d77t@jO@jFEG_EGP-5o-5`D3OD3N$6$6NM_M`1(o1(h2121j/Lo/Ld7Qo7Qt5o5~  4&:?+D 1 1h 0  0 f[:|&  &  i