A076036 G.f.: 1/(1 - 5*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

1, 5, 30, 185, 1150, 7170, 44760, 279585, 1746870, 10916150, 68219860, 426353130, 2664633580, 16653699860, 104084695500, 650526003825, 4065775405350, 25411052086350, 158818913483700, 992617612224750, 6203857867325700, 38774103465635100, 242338116077385600

Offset: 0

Author

N. J. A. Sloane, Oct 29 2002

Keywords

• nonn
• easy

Comments

Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 5*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Crossrefs

Cf. A000108, A000984, A007854, A076035, A126694.

Programs

PARI

C(x) = (1 - sqrt(1 - 4*x))/(2*x);
my(x = 'x + O('x^25)); Vec(1/(1 - 5*x*C(x))) \\ Michel Marcus, Jan 21 2020

Formula

a(n) = Sum_{k = 0..n} A106566(n, k)*5^k. - Philippe Deléham, Sep 01 2005

a(n) = Sum{k = 0..n} A039599(n,k)*4^k. - Philippe Deléham, Sep 08 2007

a(0) = 1, a(n) = (25*a(n-1) - 5*A000108(n-1))/4 for n >= 1. - Philippe Deléham, Nov 27 2007

a(n) = Sum_{k = 0..n} A116395(n,k)*3^k. - Philippe Deléham, Sep 27 2009

D-finite with recurrence: +4*n*a(n) +(-41*n+24)*a(n-1) +50*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

a(n) = 5*A076025(n), n>0. - R. J. Mathar, Jan 20 2020