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A326321
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Sum of the n-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.
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4
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1, 1, 5, 271, 395793, 28076306251, 150414812114874563, 86530666539373619904011413, 7177587537701279221012034803727966465, 110824376322428312270365608303690048162629868273811, 399431453468560513224979712848478555015392084082614167438553312275
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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a(2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
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MAPLE
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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);
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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