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

0,2

Links

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

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

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 *)

Crossrefs

Sequence in context: A367818 A303271 A289802 * A370034 A289927 A164619

Adjacent sequences: A071716 A071717 A071718 * A071720 A071721 A071722

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved