bibkey: schulte2017a208342 authors: Werner Schulte year: 2017 title: “OEIS A208342, Triangle of coefficients of polynomials u(n,x) jointly generated with A208343” doi: null url: https://oeis.org/A208342 claim: “%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section. %F u(n,x) = u(n-1,x) + xv(n-1,x), %F v(n,x) = xu(n-1,x) + x*v(n-1,x), %F where u(1,x) = 1, v(1,x) = 1. %F T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - Werner Schulte, Mar 07 2017” strata_touched:

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

OEIS A208342

The entry defines a jointly generated polynomial pair and records the coefficient triangle of the first polynomial. Its NAME and defining FORMULA lines are reproduced in the claim above. The coefficient convention is T(n,k) = [x^(k-1)] u(n,x).

The final quoted FORMULA line attributes to Werner Schulte the conjectured binomial-sum expression for every positive k,n with k<=n. The formal module uses the coefficient recursion induced by the displayed polynomial recursion and proves that expression for the stated range.

Verified locator

• URL: https://oeis.org/A208342
• Locator: FORMULA, Werner Schulte, Mar 07 2017.

T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - Werner Schulte, Mar 07 2017