A071715 Expansion of (1+x*C^4)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 5, 22, 92, 373, 1482, 5810, 22576, 87210, 335597, 1288276, 4938100, 18913245, 72416190, 277278570, 1061970720, 4069098750, 15600014370, 59845059924, 229737429048, 882572168450, 3393064828340, 13054529476852, 50264063289888, 193677360795380, 746828236103677

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

Conjecture: (891*n+6824)*(n+7)*a(n) + (891*n^2-38453*n-250970)*a(n-1) + 4*(-5040*n^2-4271*n+8095)*a(n-2) + 4*(1170*n+3847)*(2*n-1)*a(n-3) = 0. - R. J. Mathar, Aug 25 2013

a(n) = 4*binomial(2*n+3,n)/(n+4) + 8*binomial(2*n+5,n+6)/(n+7). - Tani Akinari, Aug 06 2025

a(n) ~ 9 * 2^(2*n+5) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 26 2025

Mathematica

a[n_] := 4 * Binomial[2*n+3, n] / (n+4) + 8 * Binomial[2*n+5, n+6] / (n+7); Array[a, 20, 0] (* Amiram Eldar, Sep 26 2025 *)

Prog

(Maxima) a(n):=4*binomial(2*n+3, n)/(n+4)+8*binomial(2*n+5, n+6)/(n+7);
makelist(a(n), n, 0, 50); /* Tani Akinari, Aug 06 2025 */

Crossrefs

Sequence in context: A172061 A211973 A053297 * A278472 A010036 A127617

Adjacent sequences: A071712 A071713 A071714 * A071716 A071717 A071718

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved