An explicit generating function arising in counting binomial coefficients divisible by powers of primes

Lukas Spiegelhofer, Michael Wallner

For a prime $p$ and nonnegative integers $j$ and $n$ let ${\vartheta }_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Mor…