%I #32 Sep 08 2022 08:45:24

%S 1,1,5,9,33,73,233,569,1693,4353,12477,32985,92637,248673,690549,

%T 1869513,5158881,14033161,38587193,105246041,288818305,788939769,

%U 2162574513,5912375033,16196093881,44300854441,121311490937

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

%C Hankel transform is 4^n. - _Paul Barry_, Jan 19 2011

%H G. C. Greubel, <a href="/A116390/b116390.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%F 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

%F 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

%F a(n) ~ (1+2/sqrt(13)) * (1+2*sqrt(13))^n / 3^(n+1). - _Vaclav Kotesovec_, Feb 03 2014

%t CoefficientList[Series[1/(2*Sqrt[1-4*x^2]-x-1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 03 2014 *)

%o (PARI) my(x='x+O('x^30)); Vec(1/(2*sqrt(1-4*x^2)-x-1)) \\ _G. C. Greubel_, May 23 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2*Sqrt(1-4*x^2)-x-1) )); // _G. C. Greubel_, May 23 2019

%o (Sage) (1/(2*sqrt(1-4*x^2)-x-1)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 23 2019

%Y Row sums of number triangle A116389.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 12 2006