bibkey: wojnar2018a301420
authors: Gregory Gerard Wojnar
year: 2018
title: "OEIS A301420: positive coefficients of five-data generalized Chebyshev polynomials"
doi: null
url: https://oeis.org/A301420
claim: "The entry defines the sum of positive monomial coefficients for five data, and conjectures its rational generating function and order-fourteen recurrence."
strata_touched:
  - D5/S3/Combinatorics/PowerSeries/PositiveNewtonSums
license: citation-only
triage: anchor

Five-data positive coefficient sums

OEIS A301420, NAME and COMMENTS, defines the sum of positive coefficients in the normalized Girard–Waring polynomials for five data. The FORMULA field asserts, with the qualifier “conjectured”, the generating function

[ \frac{1-x(1+x)^4}{(1-x)(2-(1+x)^5)} ]

and the order-fourteen recurrence reproduced in the corresponding formal source. The offset is one. Interpreting the displayed generating function as the shifted series with coefficient a(k) at degree k-1, and taking the recurrence at every positive index, are explicit mathematical interpretations of that offset. The ratio assertion is a separate claim.

G. G. Wojnar, D. Sz. Wojnar and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381v2, equation (2), pp. 3–4, gives the normalized Girard–Waring coefficient. For a multiplicity vector m of weighted degree k and r factors, its five-data specialization is

[ c_m=\frac{k}{5r}(-1)^{k+r}\frac{r!}{\prod_i m_i!} \prod_{i=1}^5\binom5i^{m_i}. ]

The source coefficient is attached to each distinct monomial before substituting data values; the positive sum therefore filters these coefficients individually. The formal source takes this literal coefficient formula as its sequence definition. It does not assert a theorem about roots or silently assume a simple-root condition.

OEIS A302764, COMMENTS, also states the whole-family numerator and denominator, including the five-data case. The inspected entry supplies no proof or proof reference for that statement. The formal coefficient-to-logarithm derivation is repository-derived; global novelty of a proof is unverified. The source record supports neither a novel-answer claim nor credit for the companion six-data sequence.

Primary locators: https://arxiv.org/abs/1706.08381v2, https://oeis.org/A301420/internal, https://oeis.org/A302764/internal.

Locator

https://oeis.org/A301420