A116390 Expansion of 1/(2*sqrt(1-4*x^2)-x-1).

1, 1, 5, 9, 33, 73, 233, 569, 1693, 4353, 12477, 32985, 92637, 248673, 690549, 1869513, 5158881, 14033161, 38587193, 105246041, 288818305, 788939769, 2162574513, 5912375033, 16196093881, 44300854441, 121311490937

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Comments

Hankel transform is 4^n. - Paul Barry, Jan 19 2011

Links

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

Crossrefs

Row sums of number triangle A116389.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2*Sqrt(1-4*x^2)-x-1) )); // G. C. Greubel, May 23 2019

Mathematica

CoefficientList[Series[1/(2*Sqrt[1-4*x^2]-x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

PARI

my(x='x+O('x^30)); Vec(1/(2*sqrt(1-4*x^2)-x-1)) \\ G. C. Greubel, May 23 2019

Sage

(1/(2*sqrt(1-4*x^2)-x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 23 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..floor(n/2)} (-1)^(k-j)*C(k,j) *C(i+(j-1)/2,i)*C(j,n-2*i)*4^i.

a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*A006130(n-2*k). - Paul Barry, Jan 19 2011

Starting with offset 1, let M = an infinite tridiagonal matrix with [1,0,0,0,...] in the main diagonal and [2,1,1,1,...] in the super and subdiagonals. Let V = vector [1,0,0,0,...]. The sequence = iterates of M*V as to the leftmost column. - Gary W. Adamson, Jun 08 2011

D-finite with recurrence: -3*n*a(n) + 2*n*a(n-1) + (29*n-36)*a(n-2) + 8*(3-n)*a(n-3) + 68*(3-n)*a(n-4)=0. - R. J. Mathar, Aug 09 2012

a(n) ~ (1+2/sqrt(13)) * (1+2*sqrt(13))^n / 3^(n+1). - Vaclav Kotesovec, Feb 03 2014