bibkey: schulte2015a008287 authors: Werner Schulte year: 2015 title: “OEIS A008287, quadrinomial coefficients, with the conjectured expansion T(n,k) = Sum_{j=0..k} (-2)^j binomial(n,j) binomial(3n-2j,k-j)” doi: null url: https://oeis.org/A008287 claim: “%N Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n. %F T(n,k) = Sum_{j=0..k} (-2)^jbinomial(n,j)binomial(3n-2j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - Werner Schulte, Sep 09 2015” strata_touched:

• D5/S1/Recurrence/SchulteQuadrinomialAlternatingBinomialExpansion license: citation-only triage: anchor

OEIS A008287

Werner Schulte’s 2015 formula proposes an alternating-binomial expansion for the triangle of quadrinomial coefficients.

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• URL: https://oeis.org/A008287

%N Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n.
%F T(n,k) = Sum_{j=0..k} (-2)^j*binomial(n,j)*binomial(3*n-2*j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - _Werner Schulte_, Sep 09 2015

Readings of 2026-09-16: the OEIS entry still marks the formula (conjectured) (added 2015-09-09) with no proof line; OpenAlex quadrinomial coefficients binomial identity returned 8 hits, none about this expansion; Math.SE API returned 0 hits; GitHub code search in google-deepmind/formal-conjectures returned 0 hits; the arXiv API gave no response; the entry’s linked Fahssi 2012 (arXiv:1202.0228) predates the conjecture and was not checked page by page (ASSUMED-UNVERIFIED). The factorisation 1+x+x²+x³ = (1+x)³ − 2x(1+x) and the binomial theorem are textbook facts; the named expansion itself was not found in these surfaces. Historical openness beyond the listed surfaces is ASSUMED-UNVERIFIED.