A165431 A transform of the central binomial coefficients.

1, 2, 6, 16, 46, 132, 388, 1152, 3462, 10492, 32036, 98400, 303756, 941576, 2928936, 9138176, 28584006, 89609196, 281466916, 885620576, 2790812196, 8806560056, 27823745016, 88005102336, 278637450396, 883024243032, 2800748951208

Offset: 0

Author

Paul Barry, Sep 18 2009

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A026569.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(8*x^3-4*x+1))); // G. C. Greubel, Oct 20 2018

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 *)

PARI

x='x+O('x^30); Vec(1/sqrt(8*x^3-4*x+1)) \\ G. C. Greubel, Oct 20 2018

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