Multinomial - Sum of coeffecients with even powers

Question

Let P be a polynomial given by $P(x_1,x_2,x_3, \ldots,x_n) = (k+x_1+x_2+\ldots +x_n)^m$.

Find the sum of all coefficients of the terms of the polynomial which have even powers in each of the $n$ variables for $n=6, m=6, k=6$.

I got answer as $83,887$ using multinomial expansion in Python and using RegEx to remove terms with odd powers, but it's incorrect. Any help will be appreciated.

Answer 1

This is not an answer to the combinatorial question, but it can help to check whether a combinatorial answer could be correct.

The following code constructs the polynomial via python's sympy and loops through the coefficients. Here p.coeff() is a list of all coefficients. p.monoms is a list of tuples with the degree of each variable. sum([m % 2 for m in monom]) == 0]) is a way to select all tuples that have only even terms.

from sympy import symbols, poly

k = 6
m = 6
n = 6
x = [symbols(f'x{i}') for i in range(1, n + 1)]
p = poly((k + sum(x)) ** m)
print(sum([coeff for coeff, monom in zip(p.coeffs(), p.monoms()) if sum([m % 2 for m in monom]) == 0]))

The result is 217392.