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Sum of n-th powers of coefficients in full expansion of (z_1 + z_2 + ... + z_n)^n.
3

%I #13 Jan 27 2023 20:53:05

%S 1,1,6,381,591460,41262262505,207874071367118436,

%T 110807909819808911886548575,8558639841332633529404511878004186120,

%U 124773193097402414339622625011223384066643153613969,431220070110830123225191271755402469908417673177630594034899052340

%N Sum of n-th powers of coefficients in full expansion of (z_1 + z_2 + ... + z_n)^n.

%H Alois P. Heinz, <a href="/A245398/b245398.txt">Table of n, a(n) for n = 0..30</a>

%F a(n) = [x^n] (n!)^n * (Sum_{j=0..n} x^j/(j!)^n)^n.

%F a(n) = A245397(n,n).

%p b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,

%p add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))

%p end:

%p a:= n-> b(n$3):

%p seq(a(n), n=0..12);

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,

%t Sum[b[n-j, i-1, k]*Binomial[n, j]^(k-1)/j!, {j, 0, n}]]];

%t a[n_] := n!*b[n, n, n];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Jun 27 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A245397.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 21 2014