A116391 Expansion of 1/((1+x)*(sqrt(1-4*x^2)-x)).

1, 0, 3, 2, 11, 14, 47, 78, 217, 408, 1039, 2086, 5065, 10560, 24931, 53194, 123403, 267222, 612903, 1340222, 3050679, 6714946, 15205967, 33622158, 75864835, 168275790, 378743151, 841959974, 1891648931, 4211866694, 9450828951

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Links

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

Crossrefs

Diagonal sums of A116389.

Programs

Magma

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

Mathematica

CoefficientList[Series[1/((1+x)(Sqrt[1-4(x^2) ]-x)),{x,0,40}],x] (* Harvey P. Dale, Sep 25 2018 *)

PARI

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

Sage

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

Formula

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

Conjecture D-finite with recurrence: n*a(n) +(n)*a(n-1) +3*(-3*n+4)*a(n-2) +3*(-3*n+4)*a(n-3) +20*(n-3)*a(n-4) +20*(n-3)*a(n-5)=0. - R. J. Mathar, Jan 23 2020

a(n) ~ 5^(n/2)/(1+sqrt(5)). - Vaclav Kotesovec, Nov 19 2021