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A326321 Sum of the n-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.

%I A326321 #24 Dec 03 2020 11:48:20
%S A326321 1,1,5,271,395793,28076306251,150414812114874563,
%T A326321 86530666539373619904011413,7177587537701279221012034803727966465,
%U A326321 110824376322428312270365608303690048162629868273811,399431453468560513224979712848478555015392084082614167438553312275
%N A326321 Sum of the n-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.
%H A326321 Alois P. Heinz, <a href="/A326321/b326321.txt">Table of n, a(n) for n = 0..30</a>
%H A326321 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%F A326321 From _Vaclav Kotesovec_, Sep 14 2019: (Start)
%F A326321 a(n) ~ (n!)^n.
%F A326321 a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). (End)
%F A326321 a(n) = (n!)^n * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^n). - _Ilya Gutkovskiy_, Jul 11 2020
%e A326321 a(2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
%p A326321 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A326321       add(b(n-i, k)/i!^k, i=1..n))
%p A326321     end:
%p A326321 a:= n-> n!^n*b(n$2):
%p A326321 seq(a(n), n=0..12);
%p A326321 # second Maple program:
%p A326321 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A326321       add(binomial(n, j)^k*b(j, k), j=0..n-1))
%p A326321     end:
%p A326321 a:= n-> b(n$2):
%p A326321 seq(a(n), n=0..10);
%t A326321 b[n_, k_] := b[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k b[j, k], {j, 0, n-1}]];
%t A326321 a[n_] := b[n, n];
%t A326321 a /@ Range[0, 10] (* _Jean-François Alcover_, Dec 03 2020, after 2nd Maple program *)
%Y A326321 Main diagonal of A326322.
%Y A326321 Cf. A215910.
%K A326321 nonn
%O A326321 0,3
%A A326321 _Alois P. Heinz_, Sep 11 2019