OFFSET
0,2
COMMENTS
Hankel transform is 2^n.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv:1310.2449 [cs.DM], 2013 (last line of text).
FORMULA
G.f.: 1/(1-2x-2x^2/(1-x^2/(1-2x-x^2/(1-x^2/(1-2x-x^2/(1-x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n-k,k)*2^(n-2k)*C(2k,k).
From Vaclav Kotesovec, Jul 28 2016: (Start)
D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) - 4*(2*n - 3)*a(n-3).
G.f.: 1/sqrt(8*x^3-4*x+1).
a(n) ~ sqrt(1 + 2/sqrt(5)) * (1+sqrt(5))^n / sqrt(Pi*n).
(End)
a(n) = 2^n*hypergeom([1/2, 1/2-n/2, -n/2],[1, -n],-4) for n>=1. - Peter Luschny, Jul 28 2016
MAPLE
a := n -> `if`(n=0, 1, 2^n*hypergeom([1/2, 1/2-n/2, -n/2], [1, -n], -4)):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Jul 28 2016
MATHEMATICA
Table[Sum[Binomial[n-k, k]*2^(n-2*k)*Binomial[2*k, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 28 2016 *)
CoefficientList[Series[1/Sqrt[8*x^3-4*x+1], {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 28 2016 *)
PROG
(PARI) x='x+O('x^30); Vec(1/sqrt(8*x^3-4*x+1)) \\ G. C. Greubel, Oct 20 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(8*x^3-4*x+1))); // G. C. Greubel, Oct 20 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 18 2009
STATUS
approved