An explicit algebraic generating function for OEIS A348410

TL;DR

This paper derives an explicit algebraic generating function for OEIS sequence A348410, confirming its D-finiteness and providing tools for further recurrence and differential equation analysis.

Contribution

It applies Lagrange-Bürmann inversion and resultants to obtain an explicit algebraic equation for the sequence's generating function, advancing understanding of its recurrence properties.

Findings

Derived an explicit algebraic equation for the generating function.

Confirmed the sequence's D-finiteness via algebraic equation.

Numerical verification of Kotesovec's recurrence for n=3 to 1000.

Abstract

For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each contributed a conjectured P-recursive recurrence -- Mathar's of order $4$, Kotesovec's of order $2$. We apply Lagrange-B\"urmann inversion to Bala's $[x^n]$ form to derive the parametric expression $A(t) = (1 - y^2)/(1 - y - 4 y^2)$, where $y = y(t)$ is implicit by $y(1-y)^2(1+y) = t$. Eliminating $y$ via resultant gives the explicit algebraic equation $P(t, A) = 0$ of degree $4$ in $A$ and degree $2$ in $t$. As an immediate corollary (Stanley's classical algebraic-implies-D-finite theorem), $A(t)$ is D-finite. Mathar's and Kotesovec's specific recurrences are not directly proven here; we only verify Kotesovec's order-$2$ recurrence numerically for $n = 3, \ldots, 1000$ and observe that an explicit ODE-and-recurrence extraction from $P(t, A) = 0$ via the standard Bostan-Chyzak-Salvy algebraic-to-holonomic procedure would close both conjectures. The supplementary archive contains a SymPy script which derives $P(t, A)$ and checks the numerical evidence.