A071723 Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 4, 15, 54, 192, 682, 2431, 8710, 31382, 113696, 414086, 1515516, 5571750, 20569590, 76228095, 283481670, 1057628550, 3957577800, 14849601090, 55859886420, 210622646520, 795898303668, 3013646759910, 11432740177564, 43448822603452, 165396657221152, 630597375548588

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = (Sum_{k=0..n} (k+1)*(k^2+k+1)*binomial(2*n-k,n))/(n+1). - Vladimir Kruchinin, Sep 28 2011

a(n) = (4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4). - Tani Akinari, Dec 01 2024

D-finite with recurrence 2*(n+4)*a(n) + 2*(-7*n-16)*a(n-1) + 3*(9*n+4)*a(n-2) + 6*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Jul 13 2025

a(n) ~ 11 * 4^(n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 01 2025

Maple

a := n -> (2*(2*n + 1)*(11*n^2 + 17*n + 12)*binomial(2*n, n))/((n + 1)*(n + 2)*(n + 3)*(n + 4)): seq(a(n), n = 0..25); # Peter Luschny, Dec 01 2024

Prog

(Maxima) a(n):=sum((k+1)*(k^2+k+1)*binomial(2*n-k, n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Sep 28 2011 */
(Maxima) a(n):=(4*binomial(2*n+3, n)+6*binomial(2*n+1, n+3))/(n+4); /* Tani Akinari, Dec 01 2024 */

Crossrefs

gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), A071722 (m=3), this sequence (m=4).

Cf. A000108, A070857, A099376, A390965.

Sequence in context: A291032 A006234 A094821 * A001559 A002311 A102349

Adjacent sequences: A071720 A071721 A071722 * A071724 A071725 A071726

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved