An explicit algebraic generating function for OEIS A348410
Tong Niu

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, 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 , Kotesovec's of order . We apply Lagrange-B\"urmann inversion to Bala's form to derive the parametric expression , where is implicit by . Eliminating via resultant gives the explicit algebraic equation of degree in and degree in . As an immediate corollary (Stanley's classical algebraic-implies-D-finite theorem), is D-finite. Mathar's and Kotesovec's specific recurrences are not directly proven here; we only verify Kotesovec's order- recurrence numerically for $n = 3,…
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