A344503 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

1, 0, -1, 3, 0, -5, 15, 0, -28, 84, 0, -165, 495, 0, -1001, 3003, 0, -6188, 18564, 0, -38760, 116280, 0, -245157, 735471, 0, -1562275, 4686825, 0, -10015005, 30045015, 0, -64512240, 193536720, 0, -417225900, 1251677700, 0, -2707475148, 8122425444, 0, -17620076360

Offset

0,4

Comments

Inverse binomial convolution of the Motzkin numbers.

Links

Table of n, a(n) for n=0..41.

Formula

a(3*n) = binomial(3*n, n) (A005809).

a(3*n - 1) = -binomial(3*n - 1, n - 1) (A025174).

a(3*n - 2) = 0.

Conjecture D-finite with recurrence -18*(2*n+1) *(2*n-1) *(n+1) *a(n) +2*(-36*n^3+554*n^2-1128*n+27) *a(n-1) +6*(-12*n^3-188*n^2+1235*n-1618) *a(n-2) +9*(54*n^3-27*n^2-183*n+320) *a(n-3) +54*(n-3) *(9*n^2-125*n+75) *a(n-4) +81 *(n-3) *(n-4) *(6*n+127) *a(n-5)=0. - R. J. Mathar, Nov 02 2021

Maple

a := n -> add((-1)^(n - k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n): seq(simplify(a(n)), n = 0..41);

Crossrefs

Cf. A064189 (Motzkin numbers), A005809, A025174, A344502.

Keyword

sign

Author

Peter Luschny, May 23 2021

Status

approved