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

0,2

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

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 *)

Crossrefs

Sequence in context: A289448 A071732 A289804 * A286955 A148938 A082306

Adjacent sequences: A071733 A071734 A071735 * A071737 A071738 A071739

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved