bibkey: hanna2025a391810
authors: Paul D. Hanna
year: 2025
title: "OEIS A391810, nonpositive Laurent coefficients and zero row sums"
doi: null
url: https://oeis.org/A391810/internal
claim: "Formula (2): zero full row sums. Formula (4): Sum_{k=0..floor((n+1)/2)^2} k^2*T(n,k)=-(2*n-5) for n >= 2. Formula (5): the negative alternating full row sum equals 2*round(2^n/3)=2*A001045(n), for n >= 1 (all conjectures)."
strata_touched:
  - D5/S3/ArithSums/HannaLaurentRowSums
  - D5/S3/ArithSums/HannaLaurentAlternatingJacobsthal
  - D5/S3/ArithSums/HannaLaurentSecondMoment
license: citation-only
triage: anchor

A391810

The entry defines an irregular triangle by the coefficients of nonpositive powers of x in the formal expansion at zero of Sum_{k=-oo..oo} x^k * (1-x^k)^(n+k), for every positive integer n. Formula (2) asks whether each full row sums to zero. The row includes the constant coefficient and ends at floor((n+1)/2)^2. Negative integer powers mean Laurent-field inverses; substituting x=1 into the infinite expression is not part of this definition.

The official internal and JSON records identify revision 39, with the formula attributed to Paul D. Hanna on December 21, 2025. The record was created on December 28, 2025; its current revision is dated February 3, 2026. The source records used for the statement have SHA256 21d3fa5293a19b145c937e8fb7f7bfcbb9f45d140dc999e9b231e79c4e522b42 (JSON) and d530cc411af4dfee97d907d590a4baa6271bfcb1a34e132fd5c0b152d80961df (internal HTML).

Formula (4): the signed second moment

The same revision explicitly poses:

(4) Sum_{k=0..floor((n+1)/2)^2} k^2 * T(n,k) = -(2*n-5) for n >= 2 (conjecture).

The official revision 39 text export has SHA256 49bbf5e8e42f91b6a3d838b2ce5c311eca58af74a83a4f6f684b09e1ddf0da83. This is the complete signed second moment, including k=0 and the inclusive natural-division bound, for every natural n>=2. T(n,k) retains its original bilateral Laurent coefficient meaning. The corresponding precise question is Problems/oeis-a391810-second-moment.md; formula (2)’s dossier remains Problems/oeis-a391810-full-laurent-row-sums.md.

The bounded source and library qualification for formula (4) is identified in issue 24520. It found no exact settlement in its checked surfaces; global solvedness, priority, complete revision history and citation coverage remain ASSUMED-UNVERIFIED. The coefficient bridge and support bound are reused from HannaLaurentRowSums; the second moment belongs to HannaLaurentSecondMoment. This does not relabel the zero-row-sum result.

Verified locator

Formula (5) retains signed coefficients, includes k=0, and uses the same inclusive upper bound floor((n+1)/2)^2. Its exact rational input to round is 2^n/3; nearest-integer ties go toward positive infinity. A001045 uses its zero offset. The revision 39 official export used for this formula has SHA256 49bbf5e8e42f91b6a3d838b2ce5c311eca58af74a83a4f6f684b09e1ddf0da83.

• URL: https://oeis.org/A391810/internal
• URL: https://oeis.org/search?q=id:A391810&fmt=json

Escape-audit boundary

The four-slot escape registration for D5.S3.ArithSums.HannaLaurentAlternatingJacobsthal.full_formula_five is unfinished under CLAUDE.md §3.9. The faithful Reg construction has not compiled, and the required proof of actual observational dependence is missing. No validated Reg mirror is retained, and mathematical compilation does not validate the registration. The concrete missing evidence is tracked in issue #24532. This is the linked-issue exception for unfinished audit work, not a judge defect.