A071719 Expansion of (1+x^2*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 4, 15, 53, 185, 647, 2277, 8073, 28834, 103700, 375326, 1366290, 4999717, 18382405, 67877025, 251615745, 936047790, 3493585920, 13077995730, 49091198550, 184742369370, 696863109774, 2634345976818, 9978753842378

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1665

Programs

Maple

f:= gfun:-rectoproc({-3*(n+4)*(3*n-1)*a(n) +5*(n+2)*(7*n+3)*a(n-1) +2*(2*n-3)*(n-7)*a(n-2)=0,a(0)=1,a(1)=4},a(n),remember):
map(f, [$0..40]); # Robert Israel, Jun 07 2018

Mathematica

(1 + c x^2) c^4 /. c -> (1 - (1 - 4x)^(1/2))/(2x) + O[x]^24 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2019 *)

Formula

Conjecture: -3*(n+4)*(3*n-1)*a(n) +5*(n+2)*(7*n+3)*a(n-1) +2*(2*n-3)*(n-7)*a(n-2)=0. - R. J. Mathar, Aug 25 2013

Conjecture confirmed using the differential equation (4*x^4+35*x^3-9*x^2)*g'' + (-14*x^3+190*x^2-42*x)*g' + (-10*x^2+150*x+12)*g = 30*x+12 satisfied by the G.f. - Robert Israel, Jun 07 2018

a(n) ~ 37 * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 23 2019