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A309999 Number of distinct values of multinomial coefficients M(n;lambda) where lambda ranges over all partitions of n into distinct parts.

%I A309999 #17 Dec 07 2020 13:59:32
%S A309999 1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,25,32,35,44,53,61,72,81,98,114,
%T A309999 130,147,176,200,229,257,291,342,387,442,501,573,642,714,807,907,1037,
%U A309999 1159,1293,1458,1624,1811,2024,2246,2505,2785,3114,3449,3795,4213,4660
%N A309999 Number of distinct values of multinomial coefficients M(n;lambda) where lambda ranges over all partitions of n into distinct parts.
%C A309999 Differs from A000009 first at n = 15: a(15) = 25 < 27 = A000009(15). There are two repeated multinomial coefficients for n = 15: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).
%H A309999 Alois P. Heinz, <a href="/A309999/b309999.txt">Table of n, a(n) for n = 0..125</a>
%H A309999 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A309999 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%p A309999 g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, {}, `if`(n=0, {1},
%p A309999      {map(x->binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]}))
%p A309999     end:
%p A309999 a:= n-> nops(g(n$2)):
%p A309999 seq(a(n), n=0..55);
%t A309999 g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Union[ Binomial[n, i] #& /@ g[n - i, Min[n - i, i - 1]], g[n, i - 1]]]];
%t A309999 a[n_] := Length[g[n, n]];
%t A309999 a /@ Range[0, 55] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A309999 Cf. A000009, A070289.
%K A309999 nonn
%O A309999 0,4
%A A309999 _Alois P. Heinz_, Aug 26 2019