U Zʹj@s4ddlmZddZddZddZefdd Zd S) )as_intcCspt|}d|fd|dfdi}d}td|ddD]8}|||d|}|||||f<||||f<q2|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 rrrangendakr X/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/ntheory/multinomial.pybinomial_coefficientss"rcCs^t|}dg|d}d}td|ddD],}|||d|}|||<|||<q,|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 rrrrr r r binomial_coefficients_listsrc Cst|}t|}|s$|riSddiS|dkr4t|S|d|krV|dkrVtt||S|gdg|d}t|di}|rd}n|}||dkr~||}|rd||<||d<|dkr||dd7<d}d}d}n,|d7}|d}|t|}||d7<t||D]@}||r||d8<||t|7}||d7<q|dd8<||||d|t|<q|S)aReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. Examples ======== >>> from sympy.ntheory import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} Notes ===== The algorithm is based on the following result: .. math:: \binom{n}{k_1, \ldots, k_m} = \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots} Code contributed to Sage by Yann Laigle-Chapuy, copied with permission of the author. See Also ======== binomial_coefficients_list, binomial_coefficients r rrr)rrdict!multinomial_coefficients_iteratortupler) mrtrjtjstartvr r r r multinomial_coefficients7sJ   rc cs\t|}t|}|d|ks$|dkr@t||}|EdHnt||}i}|D]\}}|||td|<qV|}|gdg|d}||}|td|} ||| fV|rd} n|} | |dkrX|| } | rd|| <| |d<| dkr || dd7<d} n| d7} || d7<|dd8<||}|td|} ||| fVqdS)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_tupleZmcZmc1r rrt1brrr r r rs>   rN)Zsympy.utilities.miscrrrrrrr r r r s J