def test_multinomial_coefficients(): assert multinomial_coefficients(1, 1) == {(1,): 1} assert multinomial_coefficients(1, 2) == {(2,): 1} assert multinomial_coefficients(1, 3) == {(3,): 1} assert multinomial_coefficients(2, 0) == {(0, 0): 1} assert multinomial_coefficients(2, 1) == {(0, 1): 1, (1, 0): 1} assert multinomial_coefficients(2, 2) == {(2, 0): 1, (0, 2): 1, (1, 1): 2} assert multinomial_coefficients(2, 3) == {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3} assert multinomial_coefficients(3, 1) == {(1, 0, 0): 1, (0, 1, 0): 1, (0, 0, 1): 1} assert multinomial_coefficients(3, 2) == {(0, 1, 1): 2, (0, 0, 2): 1, (1, 1, 0): 2, (0, 2, 0): 1, (1, 0, 1): 2, (2, 0, 0): 1} mc = multinomial_coefficients(3, 3) assert mc == {(2, 1, 0): 3, (0, 3, 0): 1, (1, 0, 2): 3, (0, 2, 1): 3, (0, 1, 2): 3, (3, 0, 0): 1, (2, 0, 1): 3, (1, 2, 0): 3, (1, 1, 1): 6, (0, 0, 3): 1} assert dict(multinomial_coefficients_iterator(2, 0)) == {(0, 0): 1} assert dict( multinomial_coefficients_iterator(2, 1)) == {(0, 1): 1, (1, 0): 1} assert dict(multinomial_coefficients_iterator(2, 2)) == \ {(2, 0): 1, (0, 2): 1, (1, 1): 2} assert dict(multinomial_coefficients_iterator(3, 3)) == mc it = multinomial_coefficients_iterator(7, 2) assert [next(it) for i in range(4)] == \ [((2, 0, 0, 0, 0, 0, 0), 1), ((1, 1, 0, 0, 0, 0, 0), 2), ((0, 2, 0, 0, 0, 0, 0), 1), ((1, 0, 1, 0, 0, 0, 0), 2)]
def _eval_derivative_n_times(self, s, n): # adapted from Mul._eval_derivative_n_times from sympy import Integer from sympy.ntheory.multinomial import multinomial_coefficients_iterator args = self.args m = len(args) if isinstance(n, (int, Integer)): # https://en.wikipedia.org/wiki/General_Leibniz_rule#More_than_two_factors terms = [] for kvals, c in multinomial_coefficients_iterator(m, n): p = VecMul(*[arg.diff((s, k)) for k, arg in zip(kvals, args)]) terms.append(c * p) return VecAdd(*terms) raise TypeError("Derivative order must be integer")