A240880 Expansion of g.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).

1, 1, -6, 33, -162, 666, -1836, -2079, 79542, -741474, 4907628, -24837030, 82449900, 53319060, -3741922008, 38613958497, -274566158298, 1475669401398, -5211777090564, -2356585871778, 240686500011588, -2593621485808596, 19047621883804056, -105353643788834598

Offset: 0

Author

Fung Lam, May 01 2014

Keywords

• sign
• easy

Comments

This sequence is the member (q=-3) of a class of generalized Catalan numbers (see A000108), with g.f. (1-sqrt(1-q*4*x*(1-(q-1)*x)))/(2*q*x), q<>0.

Links

• Fung Lam, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A258723.

Programs

Mathematica

CoefficientList[Series[(Sqrt[1+12x+48x^2]-1)/(6x),{x,0,30}],x]  (* Harvey P. Dale, May 24 2022 *)

Formula

G.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).

D-finite with recurrence: (n+3)*a(n+2)+6*(2*n+3)*a(n+1)+48*n*a(n)=0, a(0)=1, a(1)=1.

Lim sup n->infinity |a(n)|^(1/n) = 4*sqrt(3) = 6.9282... - Vaclav Kotesovec, May 02 2014

a(n) ~ 3^(n/2-1)*4^n / (n^(3/2)*sqrt(Pi)) * (sqrt(3)*cos(5*Pi*n/6) + 3*sin(5*Pi*n/6) - (15*sqrt(3)*cos(5*Pi*n/6) + 9*sin(5*Pi*n/6))/(8*n)). - Vaclav Kotesovec, May 02 2014