+ i0^RIHtRtRtRt]3RltR#))as_intc\V4p^V3^V^3^/p^p\^V^,^,4F9pW V, ^,,V,pV;WW, 3&WV, V3&K; V#)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 rrangendaks& y/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/ntheory/multinomial.pybinomial_coefficientsr sx" q A QQFAA A 1adQh  a%!)_q $%%QU( aAqk  Hc\V4p^.V^,,p^p\^V^,^,4F/pW V, ^,,V,pV;W&WV, &K1 V#)a#Return a list of binomial coefficients as rows of the Pascal's triangle. Examples ======== >>> from sympy.ntheory import binomial_coefficients_list >>> binomial_coefficients_list(9) [1, 9, 36, 84, 126, 126, 84, 36, 9, 1] See Also ======== binomial_coefficients, multinomial_coefficients rrs& r binomial_coefficients_listrsg q A q1u A A 1adQh  a%!)_q qQx  Hrc\V4p\V4pV'gV'd/#R^/#V^8Xd \V4#V^V,8dV^8d\\W44#V.^.V^, ,,p\ V4^/pV'd^pMTpW@^, 8EdW$,pV'd ^W$&WR^&V^8d#W$^,;;,^, uu&^p^p^pM8V^, pV^,pV\ V4,pW$;;,^, uu&\ W`4FSpW(,'gKW(;;,^,uu&Ws\ V4,, pW(;;,^, uu&KU V^;;,^,uu&Wu,W^,, ,V\ V4&EK)V#)aReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. Examples ======== >>> from sympy.ntheory import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} Notes ===== The algorithm is based on the following result: .. math:: \binom{n}{k_1, \ldots, k_m} = \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots} Code contributed to Sage by Yann Laigle-Chapuy, copied with permission of the author. See Also ======== binomial_coefficients_list, binomial_coefficients )rr dict!multinomial_coefficients_iteratortupler) mrtrjtjstartvr s && r multinomial_coefficientsr7si: q Aq A IAwAv$Q''AaCxAE5a;<< qcQUmA q1 A   !e) T ADaD 6 !eHMHAEA FAEE%( A DAIDuAtt uQx[   ! ! v1t8,%( Hrc # "\V4p\V4pV^V,8gV^8Xd&\W4pVP4RjxL R#\W4p/pVP4FwrVWdV!\RV44&K TpV.^.V^, ,,pV!V4pV!\RV44p WV ,3xV'd^p MTp W^, 8dWz,p V 'd ^Wz&W^&V ^8dWz^,;;,^, uu&^p MV ^, p Wz;;,^, uu&V^;;,^,uu&V!V4pV!\RV44p WV ,3xKR#EL55i)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) N)rritemsfilter) rr_tuplemcmc1r rrt1brrs &&& r rrsR, q Aq A1Q3w!q& %a +88: %a +HHJDA+,vdA' (  C1#Q-  AY 6$# $a5k AAa%iB!Ava%A Q  aDAIDBvdB'(A!u+ !# sA F F D5FN)sympy.utilities.miscrr rrrrrrr r's#' 4 2G T49;r