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A326321 Sum of the n-th powers of multinomials M(n; mu), where mu ranges over all compositions of n. 4
1, 1, 5, 271, 395793, 28076306251, 150414812114874563, 86530666539373619904011413, 7177587537701279221012034803727966465, 110824376322428312270365608303690048162629868273811, 399431453468560513224979712848478555015392084082614167438553312275 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..30

Wikipedia, Multinomial coefficients

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

EXAMPLE

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

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-Fran├žois Alcover, Dec 03 2020, after 2nd Maple program *)

CROSSREFS

Main diagonal of A326322.

Cf. A215910.

Sequence in context: A329610 A283518 A153322 * A234324 A066210 A036213

Adjacent sequences:  A326318 A326319 A326320 * A326322 A326323 A326324

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 11 2019

STATUS

approved

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Last modified August 15 00:08 EDT 2022. Contains 356122 sequences. (Running on oeis4.)