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

%I A325308 #21 Jul 14 2025 15:20:54
%S A325308 1,1,3,10,47,246,1602,11271,93767,847846,8618738,94966191,1149277802,
%T A325308 14946737339,210112991441,3152429219400,50538450211103,
%U A325308 859238687076542,15481605986593038,294161321911723167,5886118362589143742,123610854463260840735,2720101086040978435931
%N A325308 Sum of all distinct multinomial coefficients M(n;lambda), where lambda ranges over the partitions of n.
%C A325308 Differs from A005651 first at n = 7: a(n) = 11271 < 11481 = A005651(7).
%H A325308 Alois P. Heinz, <a href="/A325308/b325308.txt">Table of n, a(n) for n = 0..90</a>
%H A325308 Milton Abramowitz and Irene A. Stegun, <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
%H A325308 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>.
%H A325308 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>.
%p A325308 g:= proc(n, i) option remember; `if`(n=0 or i=1, {n!}, {map(x->
%p A325308       binomial(n, i)*x, g(n-i, min(n-i, i)))[], g(n, i-1)[]})
%p A325308     end:
%p A325308 a:= n-> add(i, i=g(n$2)):
%p A325308 seq(a(n), n=0..23);
%t A325308 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]]];
%t A325308 a[n_] := Total[g[n, n]];
%t A325308 a /@ Range[0, 23] (* _Jean-François Alcover_, May 06 2020, after Maple *)
%Y A325308 Column k=1 of A325305.
%Y A325308 Cf. A005651.
%K A325308 nonn
%O A325308 0,3
%A A325308 _Alois P. Heinz_, Sep 05 2019