%I #10 Aug 31 2020 04:10:06
%S 1,3,34,587,12870,337877,10262004,352436961,13465074758,565280386625,
%T 25826066397756,1274138666796217,67446164001827356,
%U 3810171540686207283,228658931521878071080,14520123059677034441895,972281769469377542763078,68443768336740463562683177
%N a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*k,k) * binomial(2*n,2*k).
%F From _Vaclav Kotesovec_, Aug 31 2020: (Start)
%F a(n) ~ (2 + sqrt(n))^(2*n + 1/2) / sqrt(8*Pi*n).
%F a(n) ~ exp(4*sqrt(n) - 4) * n^(n - 1/4) / sqrt(8*Pi) * (1 + 19/(3*sqrt(n)) + 199/(18*n)). (End)
%t a[n_] := Sum[If[n == 0, Boole[n == k], n^(n - k)] * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Aug 25 2020 *)
%o (PARI) {a(n) = sum(k=0, n, n^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}
%Y Main diagonal of A337389.
%Y Cf. A337387.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Aug 25 2020