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a(n) = Sum_{k=0..n} binomial(n, k)^2 * k^(n - k).
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%I #7 Nov 12 2023 07:51:18

%S 1,1,5,28,209,1826,18217,203106,2487361,33077566,473318201,7234847126,

%T 117435618577,2014339775800,36360190887217,688237505878726,

%U 13618646813974785,280960214041690038,6028928694559721305,134277542969681115870,3098232871805383942801

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

%F a(n) = Sum_{k=0..n} binomial(n, k) * A059297(n, k).

%F log(a(n)) ~ n*(log(n) - log(log(n)) - 1 + (3*log(log(n)) + 2)/log(n) - 1/log(n)^2). - _Vaclav Kotesovec_, Nov 12 2023

%p a := n -> add(binomial(n, k)^2*k^(n - k), k = 0 .. n):

%p seq(a(n), n = 0..22);

%t Join[{1}, Table[Sum[Binomial[n,k]^2 * k^(n-k), {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 12 2023 *)

%Y Cf. A059297.

%K nonn

%O 0,3

%A _Peter Luschny_, Nov 11 2023