+ j~^RIHt^RIHt^RIHtHtHtHt^RI H t ^RI H t H t Ht^RIHt^RIHtHtHt^RIHt^R IHt^R IHt^R IHt^R IHt^R I H!t"H#t#H$t%!RR] 4t&!RR]&4t!!RR]&4t'!RR]&4t(!RR]&4t)!RR]&4t*!RR]&4t+]*t,]+t-!RR]&4t.R#)) annotations)reduce)SsympifyDummyMod)cacheit)DefinedFunctionArgumentIndexError PoleError) fuzzy_and)IntegerpiI)Eq)gmpy)sieve) binomial_mod)Poly) factorialprodsqrtc ]tRt^tRtRtRtR#)CombinatorialFunctionz(Base class for combinatorial functions. c z^RIHpV!V4pVR,pV!V4VR,V!V4,8:dV#V#)r)combsimpmeasureratio)sympy.simplify.combsimpr)selfkwargsrexprrs&, i/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/functions/combinatorial/factorials.py_eval_simplify$CombinatorialFunction._eval_simplifys=4~# 4=F7OGDM9 9K N)__name__ __module__ __qualname____firstlineno____doc__r$__static_attributes__r'r&r#rrs 2r&rc,]tRt^$t$RtR(Rlt.^N^N^N^N^N^N^N^#N^#NRN^?NRN^NRNRNRNRNRNR NR NR NR NR NRNRNRNRNRNRNRNRNRNRNt.tR]R&] R4t ] R4t ] R4t Rt RtR)RltRtR tR!tR"tR#tR$tR%tR&tR'#)*raImplementation of factorial function over nonnegative integers. By convention (consistent with the gamma function and the binomial coefficients), factorial of a negative integer is complex infinity. The factorial is very important in combinatorics where it gives the number of ways in which `n` objects can be permuted. It also arises in calculus, probability, number theory, etc. There is strict relation of factorial with gamma function. In fact `n! = gamma(n+1)` for nonnegative integers. Rewrite of this kind is very useful in case of combinatorial simplification. Computation of the factorial is done using two algorithms. For small arguments a precomputed look up table is used. However for bigger input algorithm Prime-Swing is used. It is the fastest algorithm known and computes `n!` via prime factorization of special class of numbers, called here the 'Swing Numbers'. Examples ======== >>> from sympy import Symbol, factorial, S >>> n = Symbol('n', integer=True) >>> factorial(0) 1 >>> factorial(7) 5040 >>> factorial(-2) zoo >>> factorial(n) factorial(n) >>> factorial(2*n) factorial(2*n) >>> factorial(S(1)/2) factorial(1/2) See Also ======== factorial2, RisingFactorial, FallingFactorial c ^RIHpHpV^8XdGV!VP^,^,4V!^VP^,^,4,#\ W4h)r)gamma polygamma)'sympy.functions.special.gamma_functionsr0r1argsr )r argindexr0r1s&& r#fdifffactorial.fdiffUsHN q=1)*9Q ! q8H+II I$T4 4r&i;ii ii#iSi{/i!imiisXiUiP ioikiIi/LiSi}i#z list[int]_small_factorialsc ~V^!8dVPV,#\\V44.r2\P!^V^,4FKp^TreWd,pV^8dV^,^8Xd WT,pK(K*T^8gK:TP T4KM \P!V^,V^,^,4F*pW,^,^8XgKVP V4K, \ \P!V^,^,V^,44p\ V4pWx,#)!) _small_swingint_sqrtr primerangeappendr) clsnNprimesprimepq L_product R_products && r#_swingfactorial._swingds r6##A& &E!H rv))!QU3!1KA1uq5A:JA&q5MM!$4))!a%A:J!#q(MM%(;U--adQhA>?IV I& &r&c ~V^8d^#VPV^,4^,VPV4,#)) _recursiverH)r?r@s&&r#rLfactorial._recursives1 q5NN1a4(!+SZZ]: :r&c \V4pVP'Ed_VP'd\P#V\P Jd\P #VP 'EdVP'd\P#VPpV^8dfVP'g9^p\^^4F&pW#,pVPPV4K( VPV^, ,pM^\e\P!V4pM?\V4P!R4pVP#V4^W, ,,p\%V4#R#R#)N1)r is_Numberis_zerorOneInfinity is_Integer is_negativeComplexInfinityrDr7ranger>_gmpyfacbincountrLr)r?r@resultibitss&& r#evalfactorial.evals AJ ;;;yyyuu ajjzz!===,,,A2v"444%&F%*1b\ &  # 5 5 < ,3    c4(A1!Qi( (!- r&c VP^,P'd(VP^,P'dR#R#R#rTNr3rqrrr s&r#_eval_is_integerfactorial._eval_is_integer5 99Q< " " "tyy|'B'B'B(C "r&c VP^,P'd(VP^,P'dR#R#R#rrrs&r#_eval_is_positivefactorial._eval_is_positiverr&c VP^,pVP'd(VP'dV^, P#R#R#rNrr xs& r# _eval_is_evenfactorial._eval_is_even: IIaL << !3t<==r&r'NrcT)r(r)r*r+r,r5r:r7__annotations__ classmethodrHrLr`rmr|rrrrrrrrr-r'r&r#rr$s.`5   "$&)+-/2479=?BDHJNPV $&+-46<>FHO %'.09;DL $&y%''<;; &+&+P<"<) * *  >r&rc]tRtRtRtR#)MultiFactorialir'N)r(r)r*r+r-r'r&r#rrsr&rcr]tRtRtRt]]R44t]R4tRt Rt Rt RRlt R t R tR tR tR #) subfactorialia<The subfactorial counts the derangements of $n$ items and is defined for non-negative integers as: .. math:: !n = \begin{cases} 1 & n = 0 \\ 0 & n = 1 \\ (n-1)(!(n-1) + !(n-2)) & n > 1 \end{cases} It can also be written as ``int(round(n!/exp(1)))`` but the recursive definition with caching is implemented for this function. An interesting analytic expression is the following [2]_ .. math:: !x = \Gamma(x + 1, -1)/e which is valid for non-negative integers `x`. The above formula is not very useful in case of non-integers. `\Gamma(x + 1, -1)` is single-valued only for integral arguments `x`, elsewhere on the positive real axis it has an infinite number of branches none of which are real. References ========== .. [1] https://en.wikipedia.org/wiki/Subfactorial .. [2] https://mathworld.wolfram.com/Subfactorial.html Examples ======== >>> from sympy import subfactorial >>> from sympy.abc import n >>> subfactorial(n + 1) subfactorial(n + 1) >>> subfactorial(5) 44 See Also ======== factorial, uppergamma, sympy.utilities.iterables.generate_derangements c V'g\P#V^8Xd\P#^^r2\^V^,4FpY4^, W2,,r2K V#rc)rrSrurX)r r@z1z2r^s&& r#_evalsubfactorial._evalGsP55L !V66M1a!e_!ebg.B%Ir&c (VP'dVP'd$VP'dVPV4#V\P Jd\P #V\P Jd\P #R#R#N)rQrUrrrrNaNrT)r?rs&&r#r`subfactorial.evalTsa ===~~~#"4"4"4yy~%uu  "zz!# r&c VP^,P'd(VP^,P'dR#R#R#r)r3is_oddrrrs&r#rsubfactorial._eval_is_even^s4 99Q<   499Q<#>#>#>$? r&c VP^,P'd(VP^,P'dR#R#R#rrrs&r#rsubfactorial._eval_is_integerbrr&c ^RIHp\R4p\PV,\ V4, p\ V4V!WT^V34,#)r) summationr^)sympy.concrete.summationsrrr NegativeOner)r rr!rr^fs&&, r#_eval_rewrite_as_factorial'subfactorial._eval_rewrite_as_factorialfs?7 #J MM1 y| +~ !C[ 999r&c ^RIHp^RIHpHp\ P V^,,V!\)\,V,4,V!V^,R4,V!V^,4,V!R4,#)r)exp)r0 lowergammarp) &sympy.functions.elementary.exponentialrr2r0rrrrr)r rrr!rr0rs&&&, r#r#subfactorial._eval_rewrite_as_gammals\>O a(aRU3Y7 37B8OOa.!"%b'* *r&c X^RIHpV!V^,R4\P, #)r) uppergammarp)r2rrExp1)r rr!rs&&, r#_eval_rewrite_as_uppergamma(subfactorial._eval_rewrite_as_uppergammarsF#'2&qvv--r&c VP^,P'd(VP^,P'dR#R#R#rrrs&r#_eval_is_nonnegative!subfactorial._eval_is_nonnegativevrr&c VP^,P'd(VP^,P'dR#R#R#r)r3is_evenrrrs&r# _eval_is_oddsubfactorial._eval_is_oddzs5 99Q<   DIIaL$?$?$?%@ r&r'Nr)r(r)r*r+r,rr rr`rrrrrrrr-r'r&r#rrs['R    "": * .r&rcL]tRtRtRt]R4tRtRtRt Rt R Rlt R t R #) factorial2iaThe double factorial `n!!`, not to be confused with `(n!)!` The double factorial is defined for nonnegative integers and for odd negative integers as: .. math:: n!! = \begin{cases} 1 & n = 0 \\ n(n-2)(n-4) \cdots 1 & n\ \text{positive odd} \\ n(n-2)(n-4) \cdots 2 & n\ \text{positive even} \\ (n+2)!!/(n+2) & n\ \text{negative odd} \end{cases} References ========== .. [1] https://en.wikipedia.org/wiki/Double_factorial Examples ======== >>> from sympy import factorial2, var >>> n = var('n') >>> n n >>> factorial2(n + 1) factorial2(n + 1) >>> factorial2(5) 15 >>> factorial2(-1) 1 >>> factorial2(-5) 1/3 See Also ======== factorial, RisingFactorial, FallingFactorial c VP'dVP'g \R4hVP'dWVP'd#V^, p^V,\ V4,#\ V4\ V^, 4, #VP'd>V\P^V, ^, ,,\ V)4, #\R4hR#)z>> ">?? !!!;;;aAa4)A,.. ~ 37(;;;zzzAMMa#gq[99Jt>> from sympy import rf, Poly >>> from sympy.abc import x >>> rf(x, 0) 1 >>> rf(1, 5) 120 >>> rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x) True >>> rf(Poly(x**3, x), 2) Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain='ZZ') Rewriting is complicated unless the relationship between the arguments is known, but rising factorial can be rewritten in terms of gamma, factorial, binomial, and falling factorial. >>> from sympy import Symbol, factorial, ff, binomial, gamma >>> n = Symbol('n', integer=True, positive=True) >>> R = rf(n, n + 2) >>> for i in (rf, ff, factorial, binomial, gamma): ... R.rewrite(i) ... RisingFactorial(n, n + 2) FallingFactorial(2*n + 1, n + 2) factorial(2*n + 1)/factorial(n - 1) binomial(2*n + 1, n + 2)*factorial(n + 2) gamma(2*n + 2)/gamma(n) See Also ======== factorial, factorial2, FallingFactorial References ========== .. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. c  a\S4o\V4pS\PJgV\PJd\P#S\PJd \ V4#VP 'EdVP 'd\P#VP'dS\PJd\P#S\PJd3VP'd\P#\P#\S\4'dKSPp\V4^8wd \R4h\!V3Rl\#\%V44^4#\!V3Rl\#\%V44^4#S\PJd\P#S\PJd\P#\S\4'dcSPp\V4^8wd \R4h^\!V3Rl\#^\'\%V44^,4^4, #^\!V3Rl\#^\'\%V44^,4^4, #VP(R8Xd9SP('d%SP*'d\P,#R#R#R#)rd0rf only defined for polynomials on one generatorc4<VSPV4,#rshiftrr^rs&&r#&RisingFactorial.eval..Qs./nr&c$<VSV,,#rr'rs&&r#rrU q!a%yr&c6<VSPV)4,#rrrs&&r#rrds01177A2;r&c$<VSV, ,#rr'rs&&r#rrhs,-q1uIr&FN)rrrrSrrUrRrrTNegativeInfinityr isinstancergenslenrrrXr;rsrqrVrur?rrrs&f& r#r`RisingFactorial.eval5s, AJ AJ :aee55L !%%ZQ<  \\\yyyuu ===AJJ zz)a000888#$#5#55#$::-%a..#$66D"4y1}&02K'L!L(./=.3CFmQ(@!@$**@*/A-$<<AJJ zz)a000 zz)%a..#$66D"4y1}&02K'L!L()1@05aSVq0I1*N(N!N$%V-6,1!SQ[1_,Eq&J$JJ <<5 ||| vv !.| !r&c ^RIHp^RIHpV'gmV^8*R8XdH\P V,V!^V, 4,V!V)V, ^,4, #V!W,4V!V4, #V!V!W,4V!V4, V^83\P V,V!^V, 4,V!V)V, ^,4, R34#rrrTrrr2r0rrr rrrr!rr0s&&&&, r#r&RisingFactorial._eval_rewrite_as_gammapsBAQ4}}a'a!e 4uaR!VaZ7HHH<%(* * 15\E!H $a!e , ]]A eAEl *UA26A:-> > EG Gr&c 4\W,^, V4#rc)FallingFactorialr rrr!s&&&,r#!_eval_rewrite_as_FallingFactorial1RisingFactorial._eval_rewrite_as_FallingFactorial{s 1--r&c T^RIHpVP'dVP'd|V!\W!,^, 4\V^, 4, V^83\P V,\V)4,\V)V, 4, R34#R#R#rrTNrrrqrrrr rrr!rs&&&, r#r*RisingFactorial._eval_rewrite_as_factorial~s}B <<!)QB-/ 1"q&0AA4HJ J);z88t G.J9C8r&rcV]tRtRtRt]R4tR RltRtRt Rt R R lt R t R t R#)r iap Falling factorial (related to rising factorial) is a double valued function arising in concrete mathematics, hypergeometric functions and series expansions. It is defined by .. math:: \texttt{ff(x, k)} = (x)_k = x \cdot (x-1) \cdots (x-k+1) where `x` can be arbitrary expression and `k` is an integer. For more information check "Concrete mathematics" by Graham, pp. 66 or [1]_. When `x` is a `~.Poly` instance of degree $\ge 1$ with single variable, `(x)_k = x(y) \cdot x(y-1) \cdots x(y-k+1)`, where `y` is the variable of `x`. This is as described in >>> from sympy import ff, Poly, Symbol >>> from sympy.abc import x >>> n = Symbol('n', integer=True) >>> ff(x, 0) 1 >>> ff(5, 5) 120 >>> ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4) True >>> ff(Poly(x**2, x), 2) Poly(x**4 - 2*x**3 + x**2, x, domain='ZZ') >>> ff(n, n) factorial(n) Rewriting is complicated unless the relationship between the arguments is known, but falling factorial can be rewritten in terms of gamma, factorial and binomial and rising factorial. >>> from sympy import factorial, rf, gamma, binomial, Symbol >>> n = Symbol('n', integer=True, positive=True) >>> F = ff(n, n - 2) >>> for i in (rf, ff, factorial, binomial, gamma): ... F.rewrite(i) ... RisingFactorial(3, n - 2) FallingFactorial(n, n - 2) factorial(n)/2 binomial(n, n - 2)*factorial(n - 2) gamma(n + 1)/2 See Also ======== factorial, factorial2, RisingFactorial References ========== .. [1] https://mathworld.wolfram.com/FallingFactorial.html .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. c  Ja\S4o\V4pS\PJgV\PJd\P#VP'dSV8Xd \ S4#VP 'EdVP 'd\P#VP'dS\PJd\P#S\PJd3VP'd\P#\P#\S\4'dKSPp\V4^8wd \!R4h\#V3Rl\%\'V44^4#\#V3Rl\%\'V44^4#S\PJd\P#S\PJd\P#\S\4'dcSPp\V4^8wd \!R4h^\#V3Rl\%^\)\'V44^,4^4, #^\#V3Rl\%^\)\'V44^,4^4, #R#)rdz0ff only defined for polynomials on one generatorc6<VSPV)4,#rrrs&&r#r'FallingFactorial.eval..s./!or&c$<VSV, ,#rr'rs&&r#rr-rr&rc4<VSPV4,#rrrs&&r#rr-s011771:r&c$<VSV,,#rr'rs&&r#rr- s AEr&N)rrrrqrrUrRrSrrTrrrrrrrrrXr;rsrs&f& r#r`FallingFactorial.evals AJ AJ :aee55L \\\a1fQ<  \\\yyyuu ===AJJ zz)a000888#$#5#55#$::-%a..#$66D"4y1}&02K'L!L(./>.3CFmQ(@!@$**@*/A-$<<AJJ zz)a000 zz)%a..#$66D"4y1}&02K'L!L()1?05aSVq0I1*N(N!N$%V,B,1!SQ[1_,Eq&J$JJSr&c ^RIHp^RIHpV'glV^8R8Xd9\P V,V!W!, 4,V!V)4, #V!V^,4V!W, ^,4, #V!V!V^,4V!W, ^,4, V^83\P V,V!W!, 4,V!V)4, R34#rr r s&&&&, r#r'FallingFactorial._eval_rewrite_as_gamma sBAA$}}a'ae 4uaRy@@Q<% "22 2 1q5\E!%!), ,a1f 5 ]]A eAEl *UA2Y 6 =? ?r&c 4\W, ^,V4#rc)rfrs&&&,r# _eval_rewrite_as_RisingFactorial1FallingFactorial._eval_rewrite_as_RisingFactorials!%!)Qr&c `VP'd\V4\W4,#R#rrrs&&&,r#r*FallingFactorial._eval_rewrite_as_binomials# <<<Q<(1.0 0 r&c T^RIHpVP'dVP'd|V!\V4\V)V,4, V^83\P V,\W!, ^, 4,\V)^, 4, R34#R#R#rrrs&&&, r#r+FallingFactorial._eval_rewrite_as_factorials}B <<>> from sympy import Symbol, Rational, binomial, expand_func >>> n = Symbol('n', integer=True, positive=True) >>> binomial(15, 8) 6435 >>> binomial(n, -1) 0 Rows of Pascal's triangle can be generated with the binomial function: >>> for N in range(8): ... print([binomial(N, i) for i in range(N + 1)]) ... [1] [1, 1] [1, 2, 1] [1, 3, 3, 1] [1, 4, 6, 4, 1] [1, 5, 10, 10, 5, 1] [1, 6, 15, 20, 15, 6, 1] [1, 7, 21, 35, 35, 21, 7, 1] As can a given diagonal, e.g. the 4th diagonal: >>> N = -4 >>> [binomial(N, i) for i in range(1 - N)] [1, -4, 10, -20, 35] >>> binomial(Rational(5, 4), 3) -5/128 >>> binomial(Rational(-5, 4), 3) -195/128 >>> binomial(n, 3) binomial(n, 3) >>> binomial(n, 3).expand(func=True) n**3/6 - n**2/2 + n/3 >>> expand_func(binomial(n, 3)) n*(n - 2)*(n - 1)/6 In many cases, we can also compute binomial coefficients modulo a prime p quickly using Lucas' Theorem [2]_, though we need to include `evaluate=False` to postpone evaluation: >>> from sympy import Mod >>> Mod(binomial(156675, 4433, evaluate=False), 10**5 + 3) 28625 Using a generalisation of Lucas's Theorem given by Granville [3]_, we can extend this to arbitrary n: >>> Mod(binomial(10**18, 10**12, evaluate=False), (10**5 + 3)**2) 3744312326 References ========== .. [1] https://www.johndcook.com/blog/binomial_coefficients/ .. [2] https://en.wikipedia.org/wiki/Lucas%27s_theorem .. [3] Binomial coefficients modulo prime powers, Andrew Granville, Available: https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf c d^RIHpV^8XdJVPwr4\W44V!^V^,4V!^W4, ^,4, ,#V^8XdJVPwr4\W44V!^W4, ^,4V!^V^,4, ,#\ W4h)r)r1)r2r1r3rr )r r4r1r@rs&& r#r5binomial.fdiffsE q=99DAA>9QA#6!QUQY'$() ) ]99DAA>9Q #:!QU#$$% %%T4 4r&c DVP'Ed VP'dV^8d\V4\V4r!W!8d\P#W!^,8d W, p\e \ \P !W44#W, ^rC\^V^,4FpV^, pWC,V,pK \ V4#W, ^rC\^V^,4FpV^, pWC,pK V\V4, #R#r) rUr;rrurYrbincoefrX _factorial)r r@rryr]r^s&&& r#rbinomial._evals <<<|||Q1vs1v1566MaZA $"5==#677E16q!a%AFA#Z1_F)v&E16q!a%AFAKF) 1 --3 r&c \\W34wrW, pVPVPrTVP'g!V'gVRJd#VP'd\ P #V^, P'g(V'gVRJdV^, P'dV#VP'dVP'g#V'd+V'd#VP'd\ P#VP'd.VPW4pV'dVPRR7#T#R#VRJdV'd\ P#VP'dD^RI HpV!V^,4V!V^,4V!W, ^,4,, #R#)FT)basicrN)rwrrrrqrRrrSrVru is_numberrexpandrWr2r0)r?r@rryn_nonnegn_isintrgr0s&&& r#r` binomial.evals$7QF# E,,all' 999(g&6III55L E???Gu,<UOOOH <<<}}}g!---vv iio14szzz-=#= 7$$ $ [[[ EQ<q1ueAEAI.>!>? ?r&c VPwr#\;QJdRW#V34F 'gK RM RM!RW#V344'd \R4h\;QJdRW#V34F 'dK RM RM!RW#V344'Ed\ \ W#34wr#\ V4^rTV^8d\P#V^8d%V)V,^, pV^,'dRM^pW28d\P#VPp\ V4pV'Ed)WB8dLY#rV'g V'd8V\Wt,W,4,V,pWt,W,rKGEMW#, p W98dYr^p \^V^,4Fp W,V,p K T p \V^,V ^,4Fp W,V,p K W\,p\V ^,V^,4Fp W[,V,pK V\W,V,V^, V4,pWT,pEM"\V4V8dV^8wd\W#V4pM\ \V44p \P !^V^,4FpWV, 8dW^,V,pK!W^,8dK0W8d'W.,W>,8dW^,V,pKZK\Y#r^;ppV^8d<\ W~,W,V,84pW~,W,rVV, pKBV^8gKV\WV4,pWT,pK \WQ,4#R#)c3<"TFqPRJxK R#5i)FN)rq.0rs& r# %binomial._eval_Mod..s8i||u$isTFz"Integers expected for binomial Modc38"TFqPxK R#5ir)rUrPs& r#rRrSs/Y||YsNrp)r3anyrallrwr;rsrrurvrrXrer<rrr=)r rEr@rrxrgrzrAKrykfr^dfMrCras&& r#r|binomial._eval_Modsyy 38qQi83338qQi8 8 8AB B 3/aAY/333/aAY/ / /sQF#DA!fa1uvv 1uBFQJaCCbQuvv kkGRBw6qq!(1616"::R? w1 ! Au 1B"1a!e_TBY-B"1q5!a%0TBY1IC"1q5!a%0!ebj13rurz26266CICqA!q&"1+aM"--aQ7E1u}!i"na 9qy0"%)b.C1 !1"# a!e #QY19q=$A BA#$:qzq1HC73u2#66CIC'8*SW: W 0r&c VVP^,pVP'd\VP!#VP^,pW#, P'd W#, pVP'dVP'd\ P #VP'd\ P#VP^,^rB\^V^,4FpWBV, V,,pK V\V4, #\VP!#)z Function to expand binomial(n, k) when m is positive integer Also, n is self.args[0] and k is self.args[1] while using binomial(n, k) ) r3rQrrUrRrrSrVrurXrE)r hintsr@rr]r^s&, r#_eval_expand_funcbinomial._eval_expand_func3s IIaL ;;;TYY' ' IIaL C   A <<<yyyuu vv  IIaL!6q!a%A!eai'F) 1 --TYY' 'r&c d\V4\V4\W, 4,, #r)rr r@rr!s&&&,r#r#binomial._eval_rewrite_as_factorialNs!|Yq\)AE*::;;r&c ^RIHpV!V^,4V!V^,4V!W, ^,4,, #rr)r r@rrr!r0s&&&&, r#rbinomial._eval_rewrite_as_gammaQs0AQU|U1q5\% *::;;r&Nc BVPW4PR4#)r)rr)r r@rr!r!s&&&&,r#r##binomial._eval_rewrite_as_tractableUs**1088EEr&c `VP'd\W4\V4, #R#r)rqffrrbs&&&,r#r*binomial._eval_rewrite_as_FallingFactorialXs# <<<a8il* * r&c VPwrVP'dVP'dR#VPRJdR#R#TFN)r3rqr r@rs& r#rbinomial._eval_is_integer\s5yy <<rs"--$NN&--$-6&== O &s>%s>j * _(_Dp0&p0p^8+^8BY8,Y8xqR$qRr&