Otj.ddlmZdZdZdZefdZdS))as_intct|}d|fd|dfdi}d}td|dzdzD]$}|||z dzz|z}|x||||z f<|||z |f<%|S)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 W/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/ntheory/multinomial.pybinomial_coefficientsrs" q A QQFAA A 1adQh  && !a%!)_q $%%!QU( aAqkk Hct|}dg|dzz}d}td|dzdzD]}|||z dzz|z}|x||<|||z <|S)aL Return a list of binomial coefficients as rows of the Pascal's triangle. Examples ======== >>> from sympy.ntheory import binomial_coefficients_list >>> binomial_coefficients_list(9) [1, 9, 36, 84, 126, 126, 84, 36, 9, 1] See Also ======== binomial_coefficients, multinomial_coefficients rrrr s rbinomial_coefficients_listrsw q A q1u A A 1adQh   !a%!)_q !qQxx Hrct|}t|}|s|riSddiS|dkrt|S|d|zkr#|dkrtt||S|gdg|dz zz}t |di}|rd}n|}||dz kr||}|r d||<||d<|dkr||dzxxdz cc<d}d}d}n/|dz }|dz}|t |}||xxdz cc<t ||D]B}||r8||xxdzcc<||t |z }||xxdz cc<C|dxxdzcc<||z||dz z|t |<||dz k|S)aReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. Examples ======== >>> from sympy.ntheory import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} Notes ===== The algorithm is based on the following result: .. math:: \binom{n}{k_1, \ldots, k_m} = \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots} Code contributed to Sage by Yann Laigle-Chapuy, copied with permission of the author. See Also ======== binomial_coefficients_list, binomial_coefficients rrr)rrdict!multinomial_coefficients_iteratortupler) mr trjtjstartvr s rmultinomial_coefficientsr7s: q Aq A   IAwAvv$Q'''AaCxxAEE5a;;<<< qcQUmA q1 A   a!e)) qT  AaDAaD 66 a!eHHHMHHHAEAA FAEE%(( A aDDDAIDDDua  At ! QuQxx[ !  ! 2v1qt8,%(( 1 a!e))2 Hrc # Kt|}t|}|d|zks|dkr,t||}|Ed{VdSt||}i}|D]!\}}|||td|<"|}|gdg|dz zz}||}|td|} ||| fV|rd} n|} | |dz kr|| } | r d|| <| |d<| dkr|| dzxxdz cc<d} n| dz } || xxdz cc<|dxxdzcc<||}|td|} ||| fV| |dz kdSdS)aqmultinomial 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 rrrs, q Aq A1Q3ww!q&& %a + +88:: %a + +HHJJ - -DAq+,CvdA'' ( (  C1#Q-  VAYY F6$## $ $2a5k  AAA!a%ii1B !!Avv!a%A Q!  aDDDAIDDDBvdB''((Ar!u+   !!a%iiiiiirN)sympy.utilities.miscrrrrrrrrrr)sk''''''   4   2G G G T49;;;;;;r