A071735 Expansion of (1+x^3*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 2, 5, 15, 47, 152, 504, 1705, 5863, 20436, 72046, 256462, 920550, 3328192, 12109270, 44305245, 162911415, 601700700, 2231255070, 8304089970, 31007435970, 116130446640, 436137803400, 1642112017338, 6197239974582, 23438771087272, 88826989017004, 337262603824860

Offset

0,2

Links

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

Formula

a(n) = (2*binomial(2*n+1,n)+5*binomial(2*n-2,n+1))/(n+2) for n>0. - Tani Akinari, Jul 24 2025

a(n) ~ 21 * 4^(n-1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 26 2025

Mathematica

a[n_] := (2*Binomial[2*n + 1, n] + 5*Binomial[2*n - 2, n + 1])/(n + 2); a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Sep 26 2025 *)

Prog

(Maxima) a(n):=if n=0 then 1 else (2*binomial(2*n+1, n)+5*binomial(2*n-2, n+1))/(n+2); makelist(a(n), n, 0, 20);  /* Tani Akinari, Jul 24 2025 */

Crossrefs

First differences of A000782.

Sequence in context: A287275 A151280 A149914 * A391035 A148363 A365267

Adjacent sequences: A071732 A071733 A071734 * A071736 A071737 A071738

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved