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, 70459635944852, 267729052908856, 1019026177217300

Offset

0,2

Links

Table of n, a(n) for n=0..27.

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

a(n) ~ 17 * 4^n / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 04 2025

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

Mathematica

a[n_] := (3*Binomial[2*n + 2, n] + 5*Binomial[2*n, n + 2])/(n + 3); Array[a, 30, 0] (* Amiram Eldar, Oct 04 2025 *)

Prog

(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 */

Crossrefs

gf = (1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), this sequence (m=3), A071723 (m=4).

Cf. A000108.

Sequence in context: A126931 A257178 A257363 * A058987 A001558 A304824

Adjacent sequences: A071719 A071720 A071721 * A071723 A071724 A071725

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved