A337388 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*k,k) * binomial(2*n,2*k).

1, 3, 34, 587, 12870, 337877, 10262004, 352436961, 13465074758, 565280386625, 25826066397756, 1274138666796217, 67446164001827356, 3810171540686207283, 228658931521878071080, 14520123059677034441895, 972281769469377542763078, 68443768336740463562683177

Offset: 0

Author

Seiichi Manyama, Aug 25 2020

Keywords

• nonn

Crossrefs

Main diagonal of A337389.

Cf. A337387.

Programs

Mathematica

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 *)

PARI

{a(n) = sum(k=0, n, n^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}

Formula

From Vaclav Kotesovec, Aug 31 2020: (Start)

a(n) ~ (2 + sqrt(n))^(2*n + 1/2) / sqrt(8*Pi*n).

a(n) ~ exp(4*sqrt(n) - 4) * n^(n - 1/4) / sqrt(8*Pi) * (1 + 19/(3*sqrt(n)) + 199/(18*n)). (End)