slug: oeis-a077864-trinomial-odd-diagonal-rational-series bibkey: schulte2015a077864 doi: null url: https://oeis.org/A077864 triage: theorem motivation_gids:

• D5/S1/Recurrence/Invariants/TrinomialOddDiagonalRationalSeries

The trinomial odd diagonal of OEIS A077864

Problem

Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3).

Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.

a(0)=1, a(1)=2, a(2)=5, a(3)=11, a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - Philippe Deléham, Oct 25 2006

Conjecture: a(n) = Sum_{j=0..n/2} A027907(n+1-j,2*j+1), n >= 0. - Werner Schulte, Sep 29 2015

Motivation

OEIS A077864 attributes the order-four recurrence to Philippe Deléham and the odd trinomial-diagonal formula to Werner Schulte. The formal target keeps the entry’s generating-function definition and expresses the proposed identity for every natural index. The coefficient sequence is treated as rational-valued, matching the power-series construction in Lean.

Gap

The preregistration search surfaces dated 2026-09-13 were the current OEIS entry and public history, the arXiv API (A077864 OR A027907, 0), the MathOverflow API (0), Crossref (0), GitHub repository search (0; code search returned 401), and OpenAlex (429). A separate Google Scholar search found only a biological false positive and no joint hit. A proof was not found in the checked surfaces. This bounded search does not establish exhaustive coverage or publication priority.

Route

For the trinomial triangle, write G for the inverse of 1 - X^2(1+X+X^2). Finite-support truncation of the substituted geometric powers gives the coefficient at degree 2n+3 as the finite sum over the reflected exponents. Reindexing by reflection and bounding the polynomial degree removes the zero tail and leaves j <= floor(n/2).

For the rational series, the two reflected inverse equations for 1-X^2-X^3-X^4 and 1-X^2+X^3-X^4, together with their denominator product, give the odd-part identity 2·X^3·(expand 2 G') = G - rescale(-1) G. Extracting the coefficient at 2n+3 identifies the rational-series coefficient with the odd diagonal sum.

Falsifier

A natural index n with a coefficient mismatch between a(n) and Sum_{j=0..n/2} A027907(n+1-j,2*j+1) would falsify the conjecture. A mismatch in the finite-support coefficient identity or in the reflected inverse equations would also invalidate the route. The exact checks through n < 60 and n <= 99 are supporting computations, not substitutes for the universal proof.

Evidence

• Lean module: D5/S1/Recurrence/Invariants/TrinomialOddDiagonalRationalSeries.lean.
• Main theorems: odd_trinomial_diagonal_coeff and schulte_a077864.
• Public definitions: trinomial, generatingSeries, and a.
• Both theorems have the standard three axioms (propext, Classical.choice, Quot.sound).
• The orchestrator’s exact check found no mismatch for n < 60.
• The probe check found no mismatch for n <= 99 and used SymPy for exact simplification of the odd-part generating function.

Triage

theorem. The Lean proof establishes the universal odd-diagonal identity and the resulting Schulte formula for the rational generating series.

ASSUMED-UNVERIFIED

The OEIS quotations, author attributions, dates, and the identification of the two entries were supplied by the preregistration and literature-search seats. The search scope is the dated set of surfaces listed above; no exhaustive literature or priority claim follows. The finite exact checks and SymPy simplification are supporting evidence supplied by the orchestrator and probe seat. The translation from the OEIS notation to the Lean expressions is not itself a kernel-checked bibliographic fact.