bibkey: barry2020a331474 authors: Paul Barry year: 2020 title: “OEIS A331474, Hankel transform of A331473” doi: null url: https://oeis.org/A331474 claim: “Hankel transform of A331473. G.f.: (1 + 3x + 16x^2 - 8x^3 + 36x^4 - 12x^5 + x^6 - x^7)/(1 + 7x^2 + x^4)^2 (conjecture).” strata_touched:

• D5/S3/Constants/Moments/A331474HankelBridge
• D5/S3/Constants/Moments/A331474HankelGeneratingFunction license: citation-only triage: anchor

OEIS A331474 and its source sequence

Paul Barry submitted A331474 on January 17, 2020 as the Hankel transform of A331473. The entry has offset 0 and prints the rational generating function

(1 + 3*x + 16*x^2 - 8*x^3 + 36*x^4 - 12*x^5 + x^6 - x^7) / (1 + 7*x^2 + x^4)^2

with the explicit qualifier (conjecture).

A331473 defines its offset-zero sequence by s(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k+2,k). Therefore A331474 means the determinants H(n) = det(s(i+j)) for 0 <= i,j <= n; it does not mean a sequence separately defined by the displayed rational function.

Verified locator

• URL: https://oeis.org/A331474
• A331474 internal revision read on 2026-09-24: #9 Jan 18 2020 07:21:58.
• NAME: Hankel transform of A331473.
• FORMULA: G.f.: (1 + 3x + 16x^2 - 8x^3 + 36x^4 - 12x^5 + x^6 - x^7)/(1 + 7x^2 + x^4)^2 (conjecture).
• OFFSET: 0,2.
• AUTHOR: Paul Barry, Jan 17 2020.
• Source URL: https://oeis.org/A331473
• A331473 internal revision read on 2026-09-24: #16 Apr 27 2020 09:28:54.
• A331473 FORMULA: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k+2,k).

The current A331474 source is still revision 9, still calls the generating function a conjecture, and contains no proof or later proof reference. The A331473 revision is later because it added other formulas, including a D-finite recurrence, but its literal alternating-binomial formula remains. These are facts about the current OEIS pages, not a claim that no proof exists outside the bounded searches recorded in the problem dossier.