A071722 Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 3, 10, 33, 110, 372, 1276, 4433, 15574, 55250, 197676, 712538, 2585292, 9434830, 34610400, 127553745, 472055910, 1753616370, 6536826780, 24443315550, 91664179620, 344655239760, 1299052403688, 4907335827258, 18576824685820

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), this sequence (m=3), A071723 (m=4).

Cf. A000108.

Programs

Maple

a := n -> (17*n^2 + 13*n + 6)*binomial(2*n, n)/((n + 1)*(n + 2)*(n + 3)): seq(a(n), n = 0..24);  # Peter Luschny, Dec 01 2024

Maxima

a(n):=(3*binomial(2*n+2,n)+5*binomial(2*n,n+2))/(n+3); makelist(a(n),n,0,50);
/* Tani Akinari, Dec 01 2024 */

Formula

Conjecture: 2*(n+3)*a(n) + 4*(-3*n-4)*a(n-1) + (17*n-9)*a(n-2) + 2*(-2*n+5)*a(n-3) = 0. - R. J. Mathar, Aug 25 2013

a(n) = (3*binomial(2*n+2, n) + 5*binomial(2*n, n+2))/(n + 3). - Tani Akinari, Dec 01 2024