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

%I #15 Sep 08 2022 08:45:24

%S 1,0,3,2,11,14,47,78,217,408,1039,2086,5065,10560,24931,53194,123403,

%T 267222,612903,1340222,3050679,6714946,15205967,33622158,75864835,

%U 168275790,378743151,841959974,1891648931,4211866694,9450828951

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

%H G. C. Greubel, <a href="/A116391/b116391.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%F a(n) ~ 5^(n/2)/(1+sqrt(5)). - _Vaclav Kotesovec_, Nov 19 2021

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

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

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

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

%Y Diagonal sums of A116389.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 12 2006