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%I #34 Jan 30 2020 21:29:17
%S 2,10,53,298,1727,10207,61154,370090,2256983,13848085,85387040,
%T 528646015,3284180720,20462505850,127816245053,800143927210,
%U 5018683475087,31532297088781,198419993271440,1250291989478773,7888160383113014
%N G.f.: (6*x+2)/(sqrt(-3*x^2-6*x+1)*(4*x^2+4*x))-(2*x+1)/(2*x^2+2*x).
%F a(n) = sum(i=0..n+1, binomial(2*n-i+1,n-i+1)*sum(j=0..n+1, binomial(j,-j+i)* binomial(n+1,j)).
%F a(n) ~ sqrt(3) * (3+2*sqrt(3))^(n+1) / (2*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Nov 23 2014
%F Conjecture D-finite with recurrence: (n+1)*a(n) +(-2*n-5)*a(n-1) +3*(-8*n+7)*a(n-2) +15*(-2*n+3)*a(n-3) +9*(-n+2)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[(6 x + 2) / (Sqrt[-3 x^2 - 6 x + 1] (4 x^2 + 4 x)) - (2 x + 1) / (2 x^2 + 2 x), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 22 2014 *)
%o (Maxima)
%o a(n):=sum(binomial(2*n-i+1,n-i+1)*sum(binomial(j,-j+i)*binomial(n+1,j), j,0,n+1),i,0,n+1);
%K nonn
%O 0,1
%A _Vladimir Kruchinin_, Nov 22 2014
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