An explicit algebraic generating function for OEIS A348410
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910225280270336 |
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | 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-Bürmann 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16553 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An explicit algebraic generating function for OEIS A348410 Niu, Tong Combinatorics 05A15, 05A19, 11B83, 13P15, 33F10 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-Bürmann 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. |
| title | An explicit algebraic generating function for OEIS A348410 |
| topic | Combinatorics 05A15, 05A19, 11B83, 13P15, 33F10 |
| url | https://arxiv.org/abs/2605.16553 |