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a(n) = Sum_{k=0..n} k^n * k! * binomial(n,k)^2.
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%I #18 Feb 07 2021 00:41:40

%S 1,1,12,315,15088,1141625,124989156,18659050795,3638892086208,

%T 897534389449809,272981684150035300,100316132701760094251,

%U 43802068733570039425776,22409162143775383385763913,13274030650412266312507931652

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

%t a[0] = 1; a[n_] := Sum[k^n * k! * Binomial[n, k]^2, {k, 0, n}]; Array[a, 15, 0] (* _Amiram Eldar_, Feb 06 2021 *)

%o (PARI) a(n) = sum(k=0, n, k^n*k!*binomial(n, k)^2);

%Y Cf. A002720, A332408, A336828, A341197.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 06 2021