Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences $$S(n,m,t)=\sum_{k_1+\cdots+k_n=m}\binom{m}{k_1,\dots,k_n}^t$$ where the sum runs over non-negative integers $k_1,\dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.

QUESTION.Is it always true that $n$ divides $S(n,m,t)$?

Observe that $S(n,m,1)=n^m$.