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arxiv: 2605.16553 · v1 · pith:WIKANR3Dnew · submitted 2026-05-15 · 🧮 math.CO

An explicit algebraic generating function for OEIS A348410

Tong Niu This is my paper

Pith reviewed 2026-05-20 16:08 UTC · model grok-4.3

classification 🧮 math.CO
keywords generating functionsalgebraic generating functionsLagrange-Bürmann inversionD-finite functionsOEIS sequencesresultantrecurrence relations
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The pith

The generating function for OEIS A348410 satisfies an explicit algebraic equation of degree 4 in the function value.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper starts from a known coefficient-extraction formula for the sequence and applies Lagrange-Bürmann inversion to produce a parametric representation of its ordinary generating function A(t). It then removes the auxiliary parameter by taking a resultant, obtaining a bivariate polynomial equation P(t, A) that vanishes identically. The resulting relation is algebraic of low degree, so the generating function is D-finite by a classical theorem. A reader would care because D-finiteness supplies a systematic route to linear recurrences and differential equations for the sequence coefficients.

Core claim

Applying Lagrange-Bürmann inversion to Bala's form a(n) = [x^n] ((1-x)(1-x^2))^{-n} produces the parametric expression A(t) = (1-y^2)/(1-y-4y^2) where y satisfies y(1-y)^2(1+y)=t. Eliminating y via the resultant yields the explicit algebraic equation P(t,A)=0 of degree 4 in A and degree 2 in t. Stanley's theorem then implies that A(t) is D-finite.

What carries the argument

Lagrange-Bürmann inversion on the coefficient-extraction form, followed by resultant elimination of the auxiliary series parameter y(t).

If this is right

  • A(t) is D-finite by Stanley's algebraic-implies-D-finite theorem.
  • Kotesovec's conjectured order-2 recurrence holds at least for n up to 1000 by direct numerical check.
  • The standard Bostan-Chyzak-Salvy procedure applied to P(t,A)=0 produces an explicit linear differential equation and recurrence for the sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The low-degree algebraic relation permits systematic extraction of asymptotics for the sequence terms via singularity analysis.
  • The same inversion-plus-resultant route may succeed for other OEIS entries that admit similar single-index binomial or coefficient-extraction presentations.
  • D-finiteness supplies a practical setting in which creative-telescoping algorithms can prove summation identities involving these numbers.

Load-bearing premise

Lagrange-Bürmann inversion applies directly to the given coefficient-extraction form without further analytic conditions on the series or the branch of y(t).

What would settle it

Expand the algebraic function defined implicitly by P(t,A)=0 as a power series in t and check whether its coefficients match the known terms of A348410 to high order.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to derive an explicit algebraic generating function for OEIS A348410. Starting from Bala's form a(n) = [x^n] ((1-x)(1-x^2))^{-n}, Lagrange-Bürmann inversion yields the parametric representation A(t) = (1-y^2)/(1-y-4y^2) where y satisfies y(1-y)^2(1+y)=t. Eliminating y by resultant produces the algebraic equation P(t,A)=0 (degree 4 in A, degree 2 in t). As a corollary, A(t) is D-finite. The order-2 recurrence conjectured by Kotesovec is verified numerically for n up to 1000, with a SymPy script provided for the algebraic steps; the conjectured recurrences themselves are not proved directly.

Significance. If the derivation is valid, the explicit algebraic equation supplies a concrete closed form that immediately implies D-finiteness via Stanley's theorem and opens the door to systematic extraction of recurrences by the Bostan-Chyzak-Salvy procedure. The combination of a standard inversion technique, resultant elimination, numerical verification to n=1000, and a reproducible SymPy script constitutes a solid, checkable contribution to the study of this sequence.

major comments (1)
  1. [main derivation (following the citation of Bala's [x^n] form)] The passage from Bala's coefficient extractor to the parametric pair A(t), y(t) invokes Lagrange-Bürmann inversion but does not record the precise formal statement used nor verify the required conditions (non-vanishing derivative at the origin and selection of the branch of y(t) that reproduces the non-negative coefficients of the original series). This verification is load-bearing for the central claim that the resulting P(t,A)=0 is the correct algebraic equation for the generating function.
minor comments (2)
  1. [numerical verification paragraph] The numerical check of the order-2 recurrence is reported only as holding for n=3 to 1000; stating the exact recurrence coefficients or the precise form being tested would make the verification easier to replicate.
  2. [final paragraph] The abstract notes that an explicit ODE-and-recurrence extraction from P(t,A)=0 would close the conjectures; performing this extraction in the supplementary script (or outlining the first few steps) would strengthen the manuscript without altering its scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the contribution, and the recommendation for minor revision. The single major comment identifies a presentational gap in the derivation that we can readily address.

read point-by-point responses

  1. Referee: [main derivation (following the citation of Bala's [x^n] form)] The passage from Bala's coefficient extractor to the parametric pair A(t), y(t) invokes Lagrange-Bürmann inversion but does not record the precise formal statement used nor verify the required conditions (non-vanishing derivative at the origin and selection of the branch of y(t) that reproduces the non-negative coefficients of the original series). This verification is load-bearing for the central claim that the resulting P(t,A)=0 is the correct algebraic equation for the generating function.

    Authors: We agree that the manuscript would benefit from an explicit citation of the precise form of the Lagrange-Bürmann theorem employed and from a short verification of the hypotheses. In the revised version we will (i) state the exact theorem (the standard formal-power-series version as in Stanley, Enumerative Combinatorics, Vol. 2, Theorem 5.4.2, or the equivalent statement in Flajolet & Sedgewick, Analytic Combinatorics, Theorem B.1), (ii) confirm that the auxiliary function f(y) = y(1-y)^2(1+y) satisfies f(0)=0 and f'(0)≠0, and (iii) note that the implicit-function theorem in the ring of formal power series guarantees a unique solution y(t) with y(0)=0 whose coefficients are non-negative (as can be verified by the first few terms or by the combinatorial interpretation of the original coefficient extractor). These additions make the passage from Bala’s extractor to the parametric representation fully rigorous and thereby confirm that the resultant P(t,A)=0 is indeed the minimal algebraic equation satisfied by the generating function. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard inversion and elimination to external coefficient form

full rationale

The paper takes Bala's externally recorded coefficient extractor a(n) = [x^n] ((1-x)(1-x^2))^{-n} as given input. It then invokes the classical Lagrange-Bürmann inversion theorem to produce the parametric form A(t) = (1-y^2)/(1-y-4y^2) with y satisfying the stated implicit equation, followed by resultant elimination to obtain the explicit algebraic equation P(t,A)=0. These steps are forward algebraic manipulations using standard operations; they do not redefine the target in terms of itself, fit parameters to subsets of the output data, or rely on load-bearing self-citations. Stanley's algebraic-implies-D-finite theorem is an independent external result. The supplementary SymPy script permits direct reproduction, confirming the chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on two classical results from algebraic combinatorics and elimination theory; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Lagrange-Bürmann inversion theorem applies to the coefficient extraction form given by Bala
    Invoked to obtain the parametric expression A(t) in terms of y(t) from the [x^n] definition.
  • domain assumption The resultant of the two polynomials in y eliminates the auxiliary variable without introducing extraneous factors
    Used to produce the explicit bivariate polynomial P(t, A).

pith-pipeline@v0.9.0 · 5834 in / 1640 out tokens · 67260 ms · 2026-05-20T16:08:34.514172+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The generating function of A348410 in OEIS using the diagonal method

    math.CO 2026-05 unverdicted novelty 4.0

    Derives an algebraic equation for the generating function of OEIS sequence A348410 using the diagonal method and Gfun software.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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