Sums Involving Multinomial Coefficients

Let \(g_k(n) = \sum_{\underline{v}\in C_k(n)} \binom{n}{v} 2^{v_1v_2 + v_2v_3 + v_3v_4 + \ldots +v_{k-1}v_k}\) where \(C_k(n)\) denote the set of \(k\)-compositions of \(n\). We show that

1. \(g_k(n+p-1) \equiv g_k(n) \pmod{p}\) for all \(k,n \geq 1\), prime \(p\);
2. \(g_k(n)\) is a polynomial in \(k\) of degree \(n\) for \(k \geq n+1\);

and, moreover, that these properties hold for wider classes of functions which are sums involving multinomial coefficients.