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

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Crossrefs

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

Cf. A000108.

Programs

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

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 */

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