A243204 Expansion of 2*x/((1-sqrt(1-2*(1-sqrt(1-4*x))))*sqrt(1-2*(1-sqrt(1-4*x))) * sqrt(1-4*x)).

1, 2, 8, 35, 160, 752, 3605, 17544, 86400, 429605, 2153008, 10860720, 55086421, 280692440, 1435868960, 7369703660, 37934443008, 195748568256, 1012292239955, 5244933087000, 27220980100160, 141486701601630, 736387364237280, 3837221866576800, 20016901815607125

Offset: 0

Author

Vladimir Kruchinin, Jun 01 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A121988.

Programs

Mathematica

CoefficientList[Series[2*x / (Sqrt[1-4*x] + Sqrt[-1+2*Sqrt[1-4*x]] *Sqrt[1-4*x] + 8*x-2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 02 2014 *)

Maxima

a(n):=sum(binomial(2*k-1,k)*binomial(2*n-k-1,n-k),k,0,n);

PARI

my(x='x+O('x^50)); Vec(2*x/((1-sqrt(1-2*(1-sqrt(1-4*x))))*sqrt(1-2*(1-sqrt(1-4*x)))*sqrt(1-4*x))) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) = Sum_{k=0..n} binomial(2*k-1,k)*binomial(2*n-k-1,n-k).

G.f.: A(x) = x*F'(x)/F(x), where F(x)=x*C(x)*C(x*C(x)), C(x) is g.f. of A000108.

a(n) ~ 2^(4*n-3/2) / (sqrt(Pi*n) * 3^(n-1/2)). - Vaclav Kotesovec, Jun 02 2014