A325308 Sum of all distinct multinomial coefficients M(n;lambda), where lambda ranges over the partitions of n.

1, 1, 3, 10, 47, 246, 1602, 11271, 93767, 847846, 8618738, 94966191, 1149277802, 14946737339, 210112991441, 3152429219400, 50538450211103, 859238687076542, 15481605986593038, 294161321911723167, 5886118362589143742, 123610854463260840735, 2720101086040978435931

Offset: 0

Author

Alois P. Heinz, Sep 05 2019

Keywords

• nonn

Comments

Differs from A005651 first at n = 7: a(n) = 11271 < 11481 = A005651(7).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..90
• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
• Wikipedia, Multinomial coefficients.
• Wikipedia, Partition (number theory).

Crossrefs

Column k=1 of A325305.

Cf. A005651.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0 or i=1, {n!}, {map(x->
      binomial(n, i)*x, g(n-i, min(n-i, i)))[], g(n, i-1)[]})
    end:
a:= n-> add(i, i=g(n$2)):
seq(a(n), n=0..23);

Mathematica

g[n_, i_] := g[n, i] = If[n == 0 || i == 1, {n!}, Union[Map[Function[x, Binomial[n, i] x], g[n - i, Min[n - i, i]]], g[n, i - 1]]];
a[n_] := Total[g[n, n]];
a /@ Range[0, 23] (* Jean-François Alcover, May 06 2020, after Maple *)