bibkey: kurkov2026a107667
authors: Paul D. Hanna; Mikhail Kurkov
year: 2026
title: "OEIS A107667: Kurkov's weighted-ballot formula for Hanna's triangular matrix"
doi: null
url: https://oeis.org/history/view?seq=A107667&v=25
claim: "Conjecture: T(n+m, m) = (n+m+1) * P(n+1, n+1, m) where P(n, k, m) = P(n-1, k, m) + (n+m) * P(n, k-1, m) for 1 < k < n with P(n, 1, m) = 1, P(n, n, m) = (n+m) * P(n, n-1, m). - _Mikhail Kurkov_, Jun 02 2026"
strata_touched:
  - D5/S1/Recurrence/Algebraic/KurkovA107667WeightedBallot
license: citation-only
triage: anchor

OEIS A107667: the weighted-ballot formula

Verified locator

Source: https://oeis.org/history/view?seq=A107667&v=25

The claim is in the FORMULA field of approved revision 25 of OEIS A107667, whose entry author is Paul D. Hanna and whose formula contributor is Mikhail Kurkov. The revision identifier is %I A107667 #25 Jun 07 2026 19:02:10. This is an OEIS-only source, with no page numbers, DOI or claim-source arXiv identifier. The source text is also available at https://oeis.org/search?q=id:A107667&fmt=text.

Source statements

NAME, verbatim:

Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^2) where SHIFT_LEFT shifts each row 1 place to the left and D is the diagonal matrix {1, 2, 3, …}.

FORMULA, verbatim:

Conjecture: T(n+m, m) = (n+m+1) * P(n+1, n+1, m) where P(n, k, m) = P(n-1, k, m) + (n+m) * P(n, k-1, m) for 1 < k < n with P(n, 1, m) = 1, P(n, n, m) = (n+m) * P(n, n-1, m). - _Mikhail Kurkov_, Jun 02 2026

OFFSET, verbatim: 0,2.

The first displayed row is 1;. Thus T is zero-indexed and T(a,a)=a+1. Lower triangularity and the matrix square give T(a,b)=sum over b+1 <= j <= a of T(a,j)T(j,b+1) when b<a. Kurkov’s three-argument P is distinct from the two-argument conjugating matrix in the entry’s earlier FORMULA. Its base rule includes P(1,1,m)=1; its diagonal rule starts at row 2. Outside r>=k>=1, P is unrestricted.

Mathematical scope

The formal statement quantifies over all natural-valued tables satisfying these rules and all n,m in the natural numbers. The proof derives a parameter-sensitive first-return convolution directly from the P recurrence and then uses strong induction on the row-column difference in T. It includes n=0 and m=0. A path-set bijection and an extension to arbitrary weight sequences are not part of the formal result.

The literature screen recorded in issue #21639 found no settlement in the inspected sources. Completeness of that screen, including citing works and general theorems in unread sources, is ASSUMED-UNVERIFIED.