bibkey: mathar2012a113409 authors: R. J. Mathar; Paul Barry; G. C. Greubel year: 2012 title: “OEIS A113409, A transform of the central binomial coefficients A001405” doi: null url: https://oeis.org/A113409 claim: “Conjecture: (n+2)a(n)-2(n+1)a(n-1) +(n-4)a(n-2) +2a(n-3) +4(2-n)*a(n-4)=0. - R. J. Mathar, Nov 07 2012” strata_touched:

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

OEIS A113409

Verified locator

DOI: null

Source: https://oeis.org/A113409

Locator: FORMULA field, R. J. Mathar’s conjecture, Nov 07 2012. The entry has offset 0,3.

Definition

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).

Here n and k are nonnegative integers, C denotes the ordinary binomial coefficient, and the sum includes both endpoints.

Conjecture

Conjecture: (n+2)a(n)-2(n+1)a(n-1) +(n-4)a(n-2) +2a(n-3) +4(2-n)*a(n-4)=0. - R. J. Mathar, Nov 07 2012

The recurrence is asserted for every integer index n >= 4, so all five sequence indices are nonnegative. Its coefficients and arithmetic are interpreted in the integers.

Scope

The recurrence is the assertion addressed by the corresponding Lean module. The algebraic generating-function equation and Kotesovec’s asymptotic formula are separate assertions; they are not consequences claimed by this note.