A068770 Generalized Catalan numbers.

1, 1, 16, 264, 4480, 77952, 1386496, 25135616, 463233024, 8658673664, 163829383168, 3132565553152, 60446638866432, 1175715287400448, 23028562592268288, 453848132868898816, 8993594212565909504

Offset

0,3

Comments

a(n) = K(8,8; n)/8 with K(a,b; n) defined in a comment to A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..750

Formula

a(n) = (8^n) * p(n, -7/8) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 8*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-32*x*(1-7*x)))/(16*x).

a(n) = (4^(n-1)*14^(1/2*n+1/2)*LegendreP(n+1,2/7*14^(1/2)) - LegendreP(n,2/7*14^(1/2))*4^n*14^(1/2*n))/n for n > 0. - Mark van Hoeij, Apr 23 2010

Recurrence: (n+1)*a(n) = 224*(2-n)*a(n-2) + 16*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(1+2*sqrt(2)) * (16+4*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

Mathematica

CoefficientList[Series[(1-Sqrt[1-32*x*(1-7*x)])/(16*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Prog

(PARI) a(n) = if(n, (4^(n-1)*14^(1/2*n+1/2)*pollegendre(n+1, 2/7*14^(1/2)) - pollegendre(n, 2/7*14^(1/2))*4^n*14^(n/2))\/n, 1) \\ Charles R Greathouse IV, Mar 19 2017

Crossrefs

Cf. A000108, A068764-A068769, A068771-A068772, A025227-A025230.

Sequence in context: A240481 A180809 A240633 * A113359 A236516 A187453

Adjacent sequences: A068767 A068768 A068769 * A068771 A068772 A068773

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved