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arxiv: 2605.21255 · v1 · pith:HCJMWXVInew · submitted 2026-05-20 · 🧮 math.CO

The generating function of A348410 in OEIS using the diagonal method

Helmut Prodinger This is my paper

Pith reviewed 2026-05-21 03:12 UTC · model grok-4.3

classification 🧮 math.CO MSC 05A15
keywords generating functiondiagonal methodalgebraic equationOEIS A348410Hautus-Klarnercombinatorics
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The pith

The generating function for OEIS sequence A348410 satisfies an algebraic equation found via the diagonal method.

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

This paper establishes an algebraic equation for the ordinary generating function g(z) of the sequence A348410. The terms a_n are defined as the coefficient of x^n in the bivariate expansion of (1-x) to the power -n times (1-x^2) to the power -n. The authors apply the 1971 diagonal method of Hautus and Klarner and use the Gfun computer algebra package to compute the equation that g(z) obeys. A reader cares because an algebraic generating function typically permits derivation of recurrences, asymptotic growth rates, and other analytic properties for the sequence.

Core claim

The generating function g(z) equals the sum of a_n z^n where a_n equals the coefficient of x^n in (1-x)^{-n} (1-x^2)^{-n}, and this g(z) satisfies an algebraic equation obtained by the Hautus-Klarner diagonal method together with the Gfun software.

What carries the argument

The Hautus-Klarner diagonal method, which extracts the main diagonal from a bivariate generating function to produce an algebraic equation for the univariate generating function.

If this is right

  • The sequence A348410 possesses an algebraic ordinary generating function.
  • A linear recurrence with polynomial coefficients for a_n follows from the algebraic equation.
  • Singularity analysis of the algebraic function yields the asymptotic growth of a_n.

Where Pith is reading between the lines

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

  • The same diagonal technique could produce algebraic equations for other sequences whose terms arise as coefficients in similar bivariate expressions.
  • The minimal polynomial obtained might reveal unexpected relations between A348410 and classical combinatorial generating functions.
  • Numerical verification of the equation for large orders of z would provide independent evidence that the method was applied correctly.

Load-bearing premise

The Hautus-Klarner diagonal method applies directly to the given bivariate form (1-x)^{-n}(1-x^2)^{-n} and Gfun returns the correct algebraic equation for the resulting univariate series.

What would settle it

Compute the first twenty coefficients a_n directly from the bivariate coefficient extraction, form the partial sum of g(z), and check whether this power series satisfies the claimed algebraic equation to machine precision.

read the original abstract

$a_n=[x^n](1-x)^{-n}(1-x^2)^{-n}$ is the sequence A348410 in the Encyclopedia of Integer Sequences. Using a method from Hautus and Klarner from 1971 and the software \textsf{Gfun} we find an algebraic equation for the generating function $g(z)=\sum_na_nz^n$.

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

0 major / 1 minor

Summary. The manuscript defines the sequence A348410 by a_n = [x^n](1-x)^{-n}(1-x^2)^{-n} and applies the Hautus-Klarner diagonal method from 1971 together with the Gfun software package to derive an algebraic equation for the ordinary generating function g(z) = sum a_n z^n.

Significance. If the computed equation holds, the work supplies an explicit algebraic relation for the generating function of a diagonal-extracted sequence, which is known to be algebraic by general theory. The explicit use of Gfun constitutes a reproducible computational step that strengthens the practical value of the 1971 method for this concrete OEIS entry.

minor comments (1)
  1. The manuscript asserts that the Hautus-Klarner method and Gfun produce the algebraic equation but does not display the explicit polynomial, the precise bivariate rational function supplied to the software, or the Gfun command sequence. Including these elements would allow direct verification of the output without external software access.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript, as well as the recommendation for minor revision. The report accurately reflects the use of the Hautus-Klarner diagonal method combined with Gfun to obtain an explicit algebraic equation for the generating function of A348410.

Circularity Check

0 steps flagged

No significant circularity in the derivation

full rationale

The paper explicitly defines the sequence via the coefficient extraction a_n = [x^n](1-x)^{-n}(1-x^2)^{-n} and then invokes the external Hautus-Klarner diagonal method (1971) together with the independent Gfun software package to compute an algebraic equation satisfied by the ordinary generating function g(z). No parameters are fitted from sequence data, no term is defined in terms of the target result, and the central computation rests on a 1971 reference and external symbolic software rather than any self-citation chain or internal reduction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the 1971 diagonal method to this bivariate generating function and the reliability of Gfun output, with no new free parameters or invented entities introduced.

axioms (1)
  • domain assumption The Hautus-Klarner diagonal method applies to the bivariate generating function (1-x)^{-n}(1-x^2)^{-n} to produce an algebraic equation for the univariate generating function.
    This is the core technique invoked to reduce the coefficient extraction problem to an algebraic equation.

pith-pipeline@v0.9.0 · 5568 in / 1329 out tokens · 60825 ms · 2026-05-21T03:12:42.847325+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    M. L. J. Hautus and D. A. Klarner. The diagonal of a double power series.Duke Math. J., 38:229–235, 1971

    work page 1971

  2. [2]

    Salvy and P

    B. Salvy and P. Zimmermann. GFUN: a maple package for the manipulation of generating and holo- nomic functions in one variable. Transactions on Mathematical Software 20 (1994), no. 2, 163–177

    work page 1994

  3. [3]

    Neil J. A. Sloane and The OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2023

    work page 2023

  4. [4]

    An explicit algebraic generating function for OEIS A348410

    Tong Niu. An explicit algebraic generating function for OEIS A348410. ArXiv 2605.16553, (2026). Department of Mathematics, University of Stellenbosch 7602, Stellenbosch, South Africa and NITheCS (National Institute for Theoretical and Computational Sciences), South Africa. Email address:warrenham33@gmail.com

    work page internal anchor Pith review Pith/arXiv arXiv 2026