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A270709 a(n) = (n+1)*Sum_{k=0..(n-1)/2} (binomial(k+1,n-2*k-1)*binomial(2*k,k)/(k+1)^2).

%I A270709 #19 Apr 10 2017 18:54:06
%S A270709 0,2,3,2,5,7,14,26,51,103,209,435,910,1930,4122,8874,19227,41893,
%T A270709 91751,201839,445841,988403,2198547,4905147,10974210,24615134,
%U A270709 55341636,124694354,281525678,636802626,1442953404,3274997130,7444505615,16946749249
%N A270709 a(n) = (n+1)*Sum_{k=0..(n-1)/2} (binomial(k+1,n-2*k-1)*binomial(2*k,k)/(k+1)^2).
%H A270709 G. C. Greubel, <a href="/A270709/b270709.txt">Table of n, a(n) for n = 0..1000</a>
%F A270709 G.f.: ((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)).
%F A270709 a(n) ~ (3*r+2) * sqrt(3-4*r^2) * 2^(2*n+2) * r^(n+3) * (r+1)^(n+1) / (n^(3/2) * sqrt(Pi)), where r = 0.41964337760708... is the real root of the equation 4*r^2*(1+r) = 1. - _Vaclav Kotesovec_, Mar 22 2016
%t A270709 Table[(n+1)*Sum[Binomial[k+1,n-2*k-1] * Binomial[2*k,k] / (k+1)^2, {k,0,(n-1)/2}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 22 2016 *)
%o A270709 (Maxima)
%o A270709 makelist(coeff(taylor(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)),x,0,15),x,n),n,0,15);
%o A270709 a(n):=(n+1)*sum((binomial(k+1,n-2*k-1)*binomial(2*k,k))/(k+1)^2,k,0,(n-1)/2);
%o A270709 (PARI) x='x+O('x^200); concat(0, Vec(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)))) \\ _Altug Alkan_, Mar 22 2016
%Y A270709 Cf. A000108.
%K A270709 nonn
%O A270709 0,2
%A A270709 _Vladimir Kruchinin_, Mar 22 2016