Expansions of $\binom{pn}{p+r}$ in Shifted Binomial Bases and a Modular Symmetry Criterion

Abstract

We study the expansion of the polynomial $g_{p,r}(n) = \binom{pn}{p+r}$ (for integers $p \ge 2$ and $r \ge 1$) in the shifted binomial basis $\bigl\{\binom{n+k-1}{p+r}\bigr\}$. Using generating functions and finite differences, we obtain a closed-form formula for the expansion coefficients $B_{p,r,k}$. We then characterize when the coefficient sequence is palindromic, showing that it exhibits reflection symmetry on its support if and only if $r \equiv 1 \pmod{p}$. The proof combines an analysis of the sequence's support with the root structure of $\binom{pX}{p+r}$. Under the same congruence condition, we show that $p$ divides every coefficient. For $r=1$, the leading coefficient simplifies to $p C_p$, where $C_p$ is the $p$-th Catalan number. Finally, computations for small values of $p$ and $r$ show that the resulting coefficient sequences coincide with selected rows of $p$-decimated multinomial triangles (OEIS A027907 and A008287).