New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations
Abstract
The alternating composition of N = 2p weighted differential operators wj(x)·∂x p of strict order p on the line ℝ ∋ x is again an operator of order p; its coefficient is the universal constant c(p) times the Wronskian of the weights w1,…,wN. Lie brackets of vector fields fix c(p=1)=1; we want to find c(p geqslant 2): e.g., c(2) = 2 or c(3) = 90. Direct symbolic expansion (over |S2p| =(2p)! permutations) fails for p geqslant 4. Taking the monomials wj = xj−1 reduces the summation to the much smaller set Φp ⊆ S2p−1 subsetneq S2p of late-growing permutations. Expressing c(p) as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to c(18) = 4.881… · 10462. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is log c(p) ≃ 2p2log p −b p2 + o̅(p2) for p≫ 1, with bgeqslant 2.6744.
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BibTeX
@misc{shah2026integer,
title = {{New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations}},
author = {Kian C. Shah and Arthemy V. Kiselev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08636},
eprint = {2610.08636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08636},
}
RIS
TY - UNPB
TI - New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations
AU - Kian C. Shah
AU - Arthemy V. Kiselev
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08636
UR - https://arxiv.org/abs/2610.08636
AB - The alternating composition of N = 2p weighted differential operators w_j(x)·∂_x^ p of strict order p on the line ℝ ∋ x is again an operator of order p; its coefficient is the universal constant c(p) times the Wronskian of the weights w_1,…,w_N. Lie brackets of vector fields fix c(p=1)=1; we want to find c(p geqslant 2): e.g., c(2) = 2 or c(3) = 90. Direct symbolic expansion (over |S_2p| =(2p)! permutations) fails for p geqslant 4. Taking the monomials w_j = x^j−1 reduces the summation to the much smaller set Φ_p ⊆ S_2p−1 subsetneq S_2p of late-growing permutations. Expressing c(p) as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to c(18) = 4.881… · 10^462. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is log c(p) ≃ 2p^2log p −b p^2 + o̅(p^2) for p≫ 1, with bgeqslant 2.6744.
ER -