Metadata
bibkey: sheppard2025a381737
authors: Mike Sheppard
year: 2025
title: "Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice"
doi: null
url: https://oeis.org/A381737/internal?rev=28
claim: "Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1)."
strata_touched:
- D5/S3/Analytic/Polynomial/HermiteAbsoluteTieOrders
- D5/S3/Analytic/Polynomial/HermiteTieRecurrence
- D5/S3/Analytic/Polynomial/HermiteTieGeneratingFunction
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A381737/internal?rev=28, revision 28, Name and Formula fields, with offset 1,1. The name is “Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.” Conjecture 1 is “a(n) = 2n(n + 2) + (n + 1)*(-1)^(n+1).”
The example uses the physicists polynomial H_8 and compares every coefficient in absolute value. The assertion concerns exactly two distinct maximizing exponents, and the formula enumerates all qualifying orders with n at least one. The original coefficient formula is DLMF 18.5.13, already represented by HermiteSignedMode.physicistsHermite and its polynomial_coeff theorem.
Conjecture 2 in the same Formula field is “a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5).” The one-based offset makes its full well-formed range n at least six. The enumeration is the same complete increasing sequence of orders used in Conjecture 1.
Conjecture 3 in the same Formula field is “G.f.: (8x + 5x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).” The generating function uses the same complete one-based enumeration, with zero constant coefficient. Its formal interpretation is over the rationals; constant coefficient one of the denominator makes its inverse a formal unit inverse.
The official OEIS data mirror is https://raw.githubusercontent.com/oeis/oeisdata/main/seq/A381/A381737.seq. Its revision field is 28, the offset is 1,1, and the Name and Formula fields agree with the revision-specific source above. The checked source has 1435 bytes and SHA-256 b74ba6bf3eb9703b1512c982d2b41136c29205fc5ea9a738de97f699863edb3f. Source finite checks are agreement only, not universal proofs.