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

1, 1, 5, 271, 395793, 28076306251, 150414812114874563, 86530666539373619904011413, 7177587537701279221012034803727966465, 110824376322428312270365608303690048162629868273811, 399431453468560513224979712848478555015392084082614167438553312275

Offset: 0

Author

Alois P. Heinz, Sep 11 2019

Keywords

• nonn

Examples

			a(2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..30
• Wikipedia, Multinomial coefficients

Crossrefs

Main diagonal of A326322.

Cf. A215910.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-i, k)/i!^k, i=1..n))
    end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..12);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n, j)^k*b(j, k), j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..10);

Mathematica

b[n_, k_] := b[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k b[j, k], {j, 0, n-1}]];
a[n_] := b[n, n];
a /@ Range[0, 10] (* Jean-François Alcover, Dec 03 2020, after 2nd Maple program *)

Formula

From Vaclav Kotesovec, Sep 14 2019: (Start)

a(n) ~ (n!)^n.

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). (End)

a(n) = (n!)^n * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^n). - Ilya Gutkovskiy, Jul 11 2020