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a(n) = Sum_{k=0..n} binomial(2*k,k)/(k+1)*binomial(2*n-1,n-k).
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%I #12 Nov 19 2021 08:09:17

%S 1,2,8,35,161,768,3773,19006,97840,513264,2737121,14805805,81082383,

%T 448805300,2507310567,14120503129,80082573017,456977964520,

%U 2621830478785,15114658956625,87508451311125,508589225952740,2966098696204660

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

%F G.f.: (2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2).

%F Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-3)*a(n) -n*(101*n^2-312*n+203)*a(n-1) +(995*n^3-5570*n^2+9567*n-4984)*a(n-2) +2*(-2393*n^3+19300*n^2-50494*n+42835)*a(n-3) +4*(2*n7)*(1408*n^2-9889*n+16690)*a(n-4) -2600*(n-5)*(2*n-7)*(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Jan 27 2020

%F a(n) ~ 5^(2*n + 1/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n - 2)). - _Vaclav Kotesovec_, Nov 19 2021

%t Table[Sum[Binomial[2k,k]/(k+1) Binomial[2n-1,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Feb 06 2019 *)

%o (Maxima)

%o taylor((2*(1-sqrt(1-((1-sqrt(1-4*x))^2)/x))*(1/sqrt(1-4*x)+1)/2*x)/((1-sqrt(1-4*x))^2),x,0,27);

%Y Cf. A000108

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Dec 03 2016