A071736 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

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

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 *)

PROG

(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 */

CROSSREFS

Sequence in context: A071732 A389933 A289804 * A286955 A148938 A082306

Adjacent sequences: A071733 A071734 A071735 * A071737 A071738 A071739

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 06 2002

STATUS

approved