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A270490 a(n) = Sum_{i=0..(n+1)/2} binomial(2*i+1,i)*binomial(2*n-2*i,n)/(2*i+1).

%I A270490 #21 Jun 05 2017 19:06:23
%S A270490 1,2,7,24,87,320,1195,4504,17102,65304,250501,964480,3724996,14424504,
%T A270490 55983091,217702880,848042197,3308490496,12924954514,50553798696,
%U A270490 197948515868,775853655760,3043672637457,11950142769664,46954356540812
%N A270490 a(n) = Sum_{i=0..(n+1)/2} binomial(2*i+1,i)*binomial(2*n-2*i,n)/(2*i+1).
%H A270490 G. C. Greubel, <a href="/A270490/b270490.txt">Table of n, a(n) for n = 0..1000</a>
%F A270490 G.f.: 1/x*C(C(x)^2)/(C(x)*(1-x/(1-C(x))^2)), where C(x)=(1-sqrt(1-4*x))/2.
%F A270490 a(n) ~ 2^(2*n+1)/sqrt(Pi*n) * (1 - Gamma(3/4)/(sqrt(Pi)*n^(1/4)) + 7*sqrt(2*Pi) / (16*n^(3/4)*Gamma(3/4))). - _Vaclav Kotesovec_, Mar 18 2016
%F A270490 Conjecture: 3*n*(n-2)*(n+2)*a(n) -4*(n+1)*(8*n^2-23*n+12)*a(n-1) +16*n *(3*n-4)*(2*n-5)*a(n-2) +8*(2*n-3)*(4*n-7)*a(n-3) -64*(2*n-5)*(n-3)*(2*n-3)*a(n-4)=0. - _R. J. Mathar_, Jun 07 2016
%t A270490 Table[Sum[Binomial[2*i+1, i]*Binomial[2*n-2*i, n]/(2*i+1), {i, 0, (n+1)/2}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 18 2016 *)
%o A270490 (Maxima) a(n):=sum(binomial(2*i+1,i)*binomial(2*n-2*i,n)/(2*i+1),i,0,(n+1)/2);
%o A270490 (PARI) for(n=0,25, print1(sum(k=0,n, binomial(2*k+1,k)*binomial(2*n-2*k,n)/(2*k+1)), ", ")) \\ _G. C. Greubel_, Jun 05 2017
%Y A270490 Cf. A000108, A092392.
%K A270490 nonn
%O A270490 0,2
%A A270490 _Vladimir Kruchinin_, Mar 18 2016