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%I #21 Jun 07 2017 00:43:01
%S 1,0,5,7,27,66,195,540,1547,4408,12663,36455,105325,305046,885573,
%T 2576007,7506675,21909720,64039119,187418205,549141249,1610708190,
%U 4729015305,13896677718,40870234629,120290710176,354292339625,1044178154305
%N a(n) = ((2*n+1)/(n+1))*Sum_{j=0..n/2} binomial(n+1, j)*binomial(n-j-1, n-2*j).
%H G. C. Greubel, <a href="/A278646/b278646.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: -(2*x*sqrt(-3*x^2-2*x+1)-2*x-2)/(-3*x^3+sqrt(-3*x^2-2*x+1)*(x^2+2*x+1)-5*x^2-x+1).
%F a(n) = (2*n + 1)*3F2(1-n/2,3/2-n/2,-n; 2,2-n; 4). - _Ilya Gutkovskiy_, Nov 25 2016
%F a(n) ~ 3^(n+3/2)/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 27 2016
%t Table[(2 n + 1)/(n + 1) Sum[Binomial[n + 1, j] Binomial[n - j - 1, n - 2 j], {j, 0, Floor[n/2]}], {n, 0, 27}] (* or *)
%t CoefficientList[Series[-(2 x # - 2 x - 2)/(-3 x^3 + # (x^2 + 2 x + 1) - 5 x^2 - x + 1) &@ Sqrt[-3 x^2 - 2 x + 1], {x, 0, 27}], x] (* _Michael De Vlieger_, Nov 24 2016 *)
%o (Maxima) a(n):=((2*n+1)*sum(binomial(n+1,j)*binomial(n-j-1,n-2*j),j,0,n/2))/(n+1);
%o (PARI) a(n) = (2*n+1)*sum(j=0, n\2, binomial(n+1, j)*binomial(n-j-1, n-2*j))/(n+1); \\ _Michel Marcus_, Dec 03 2016
%Y Cf. A005043.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Nov 24 2016
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