A071736 Expansion of (1+x^3*C^3)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 3, 9, 29, 96, 324, 1111, 3861, 13572, 48178, 172482, 622098, 2258416, 8246190, 30264435, 111585765, 413126460, 1535267250, 5724840990, 21413721510, 80326153440, 302105210160, 1138957917318, 4303550907234, 16294686579016

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Comments

a(n) = number of Dyck (n+3)-paths whose initial ascent has length divisible by 3. For example, UUUUDDUDDD has initial ascent of length 4 and a(1) counts UUUDUDDD, UUUDDUDD, UUUDDDUD. - David Callan, Jul 25 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Mathematica

CoefficientList[Series[(1 + x^3 ((1 - (1 - 4 x)^(1/2))/(2 x))^3) ((1 - (1 - 4 x)^(1/2))/(2 x))^3, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

Maxima

a(n):=if n=0 then 1 else 3*binomial(2*n,n)*(5*n^2+3*n+4)/((n+1)*(n+2)*(n+3)); /* Tani Akinari, Aug 03 2025 */

Formula

a(n) ~ 15*4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

a(n) = 3*binomial(2*n,n)*(5*n^2+3*n+4)/((n+1)*(n+2)*(n+3)) for n>0. - Tani Akinari, Aug 03 2025