combinatorics

Expansions of $(p + r p n)$ in Shifted Binomial Bases and a Modular Symmetry Criterion

arXiv:2607.12173

summary

The paper derives a closed‑form expression for the coefficients when expanding the polynomial $\binom{pn}{p+r}$ in a shifted binomial basis, identifies a modular condition (r ≡ 1 mod p) that makes the coefficient sequence palindromic and divisible by p, and relates these sequences to known OEIS entries and Catalan numbers.

Abstract

We study the expansion of the polynomial $g_{p, r} (n) = (p + r p n)$ (for integers $p \geq 2$ and $r \geq 1$ ) in the shifted binomial basis ${(p + r n + k - 1)}$ . 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 $r \equiv 1 (mod p)$ . The proof combines an analysis of the sequence's support with the root structure of $(p + r pX)$ . 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).

7 pages, 1 table. Establishes structural links to OEIS A007318, A120906, A027907, and A008287. Includes a Python script for symbolic verification

Topics & keywords

#binomial coefficients #shifted binomial basis #palindromic sequences #generating functions #OEIS connectionsB_{p,r,k}Catalan numbersp‑decimated multinomial trianglesmodular symmetry criterionfinite differencesPython symbolic verification

Expansions of (p+rpn​) in Shifted Binomial Bases and a Modular Symmetry Criterion

Abstract

Topics & keywords

Expansions of $(p + r p n)$ in Shifted Binomial Bases and a Modular Symmetry Criterion