OFFSET
0,2
COMMENTS
a(n) is also the number of paths in a binary tree of length 2n+3 between two vertices that are 3 steps apart. - David Koslicki, (koslicki(AT)math.psu.edu), Nov 02 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = ((8*n+12)/(3*n+12))*((3*n+3)/(n+3))*2^n*C(n+1), where C(n) and the Catalan numbers of A000108.
Conjecture: (n+4)*a(n)-4*(3n+7)*a(n-1)+16*(2n+1)*a(n-2)=0. - R. J. Mathar, Dec 13 2011
From Benedict W. J. Irwin, Jul 12 2016: (Start)
G.f.: (1-sqrt(1-8*x)+4*x*(2*x-2+sqrt(1-8*x)))/(32*x^4).
E.g.f: E^(4*x)*(2*x*(4*x-3)*BesselI(0,4*x) + (3-4*x+ 8*x^2)* BesselI(1, 4*x))/(4*x^3). (End)
a(n) ~ 2^(3*n+5)*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
MATHEMATICA
CoefficientList[Series[(1-Sqrt[1-8z]+4z(-2+Sqrt[1-8z]+2z))/(32z^4), {z, 0, 20}], z] (* Benedict W. J. Irwin, Jul 12 2016 *)
PROG
(PARI) x='x+O('x^50); Vec((1-sqrt(1-8*x) + 4*x*(2*x-2+ sqrt(1-8*x)) )/(32*x^4)) \\ G. C. Greubel, May 24 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 08 2004
STATUS
approved