OFFSET
0,2
COMMENTS
a(n) is also the number of paths of length 2(n+1) in a binary tree between two vertices that are 2 steps apart. [David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]
FORMULA
a(n) = 6(n+1)*2^(n-2)*Cat(n+2)/(2n+3), where Cat(n)=A000108(n). - Ralf Stephan, Mar 11 2004
G.f.: c(2x)^3, where c(x) is the g.f. of A000108; a(n)=3(n+1)2^n*Cat(n+1)/(n+3); - Paul Barry, Dec 08 2004
a(n) = (n+1) * A000257(n+1). - F. Chapoton, Feb 26 2024
D-finite with recurrence: (n+3)*a(n) -2*(5*n+7)*a(n-1) +8*(2*n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2011
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 22/49 + 808*arcsin(1/(2*sqrt(2)))/(147*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 26/81 + 376*arcsinh(1/(2*sqrt(2)))/243. (End)
MATHEMATICA
CoefficientList[Series[8/(1 + Sqrt[1 - 8*x])^3, {x, 0, 21}], x] (* Amiram Eldar, Mar 24 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 13 2003
STATUS
approved