*h.SSKJr SrSrSr\4Sjrg))as_intc[U5nSU4SUS4S0nSn[SUS-S-5H!nX U- S--U-nU=XX- 4'XU- U4'M# U$)aReturn a dictionary containing pairs :math:`{(k1,k2) : C_kn}` where :math:`C_kn` are binomial coefficients and :math:`n=k1+k2`. Examples ======== >>> from sympy.ntheory import binomial_coefficients >>> binomial_coefficients(9) {(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84, (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1} See Also ======== binomial_coefficients_list, multinomial_coefficients rrrangendaks V/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/ntheory/multinomial.pybinomial_coefficientsrsx" q A QQFAA A 1adQh  a%!)_q $%%QU( aAqk  Hc[U5nS/US--nSn[SUS-S-5HnX U- S--U-nU=X'XU- 'M U$)a#Return a list of binomial coefficients as rows of the Pascal's triangle. Examples ======== >>> from sympy.ntheory import binomial_coefficients_list >>> binomial_coefficients_list(9) [1, 9, 36, 84, 126, 126, 84, 36, 9, 1] See Also ======== binomial_coefficients, multinomial_coefficients rrrr s rbinomial_coefficients_listrsg q A q1u A A 1adQh  a%!)_q qQx  Hrc[U5n[U5nU(d U(a0$SS0$US:Xa [U5$USU-:aUS:a[[X55$U/S/US- --n[ U5S0nU(aSnOUnX@S- :aX$nU(aSX$'XRS'US:aX$S-==S- ss'SnSnSnO$US- nUS-nU[ U5nX$==S- ss'[ X`5H6nX((dMX(==S-ss'Xs[ U5- nX(==S- ss'M8 US==S-ss'Xu-XS- -U[ U5'X@S- :aMU$)aReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. Examples ======== >>> from sympy.ntheory import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} Notes ===== The algorithm is based on the following result: .. math:: \binom{n}{k_1, \ldots, k_m} = \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots} Code contributed to Sage by Yann Laigle-Chapuy, copied with permission of the author. See Also ======== binomial_coefficients_list, binomial_coefficients rrr)rrdict!multinomial_coefficients_iteratortupler) mr trjtjstartvr s rmultinomial_coefficientsr7sv: q Aq A IAwAv$Q''AaCxAE5a;<< qcQUmA q1 A   !e) T ADaD 6 !eHMHAEA FAEE%( A DAIDuAtt uQx[   ! ! v1t8,%( 1 !e)2 Hrc ## [U5n[U5nUSU-:dUS:Xa$[X5nUR5ShvN g[X5n0nUR5HupVXdU"[SU55'M UnU/S/US- --nU"U5nU"[SU55n XU 4v U(aSn OUn XS- :avXzn U (aSXz'XS'U S:aXzS-==S- ss'Sn OU S- n Xz==S- ss'US==S-ss'U"U5nU"[SU55n XU 4v XS- :aMuggN7f)a1multinomial coefficient iterator This routine has been optimized for `m` large with respect to `n` by taking advantage of the fact that when the monomial tuples `t` are stripped of zeros, their coefficient is the same as that of the monomial tuples from ``multinomial_coefficients(n, n)``. Therefore, the latter coefficients are precomputed to save memory and time. >>> from sympy.ntheory.multinomial import multinomial_coefficients >>> m53, m33 = multinomial_coefficients(5,3), multinomial_coefficients(3,3) >>> m53[(0,0,0,1,2)] == m53[(0,0,1,0,2)] == m53[(1,0,2,0,0)] == m33[(0,1,2)] True Examples ======== >>> from sympy.ntheory.multinomial import multinomial_coefficients_iterator >>> it = multinomial_coefficients_iterator(20,3) >>> next(it) ((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1) rrNr)rritemsfilter) rr _tuplemcmc1r rrt1brrs rrrsX, q Aq A1Q3w!q& %a +88: %a +HHJDA+,vdA' (  C1#Q-  AY 6$# $a5k AAa%iB!Ava%A Q  aDAIDBvdB'(A!u+ !a%i# sAEEC7EEN)sympy.utilities.miscrrrrrrrrrr)s#' 4 2G T49;r