Let $p$ be a prime and $a_0,\ldots,a_n\geq 0$ be integers. Define $$ S(a_0,\ldots,a_n)=\sum_{k=0}^n a_k\binom{n}{k}. $$
I am trying to find out how much we know about $$ v_p(S(a_0,\ldots,a_n)), $$ where $v_p$ is the $p-$adic valuation, or identities involving $S(a_0,\ldots,a_n)$.
Here are a few classical ones, $S(1,x,\ldots,x^n)=(1+x)^n$ and $S(0,1,\ldots,n)=n2^{n-1}$.