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A326493 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n into distinct parts (k is a partition length).

%I A326493 #73 Jul 17 2023 12:35:39
%S A326493 1,1,1,2,2,5,9,21,38,146,322,902,3106,8406,35865,123321,393691,
%T A326493 1442688,7310744,23471306,129918661,500183094,2400722981,9592382321,
%U A326493 47764284769,280267554944,1247781159201,7620923955225,36278364107926,189688942325418,1124492015730891
%N A326493 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n into distinct parts (k is a partition length).
%C A326493 Number of partitions of [n] such that each block contains its size as an element. So the block sizes have to be distinct. a(6) = 9: 123456, 12|3456, 1345|26, 1346|25, 1456|23, 1|23456, 1|24|356, 1|25|346, 1|26|345.
%H A326493 Alois P. Heinz, <a href="/A326493/b326493.txt">Table of n, a(n) for n = 0..731</a>
%H A326493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A326493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A326493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A326493 with(combinat):
%p A326493 a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0),
%p A326493         p=select(l-> nops(l)=nops({l[]}), partition(n))):
%p A326493 seq(a(n), n=0..30);
%p A326493 # second Maple program:
%p A326493 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, p!,
%p A326493       b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)/(i-1)!))
%p A326493     end:
%p A326493 a:= n-> b(n$3):
%p A326493 seq(a(n), n=0..31);
%t A326493 b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p-1]/(i-1)!]];
%t A326493 a[n_] := b[n, n, n];
%t A326493 a /@ Range[0, 31] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A326493 Cf. A000110, A007837, A327711, A327712, A362362, A363881, A364207, A364278.
%K A326493 nonn
%O A326493 0,4
%A A326493 _Alois P. Heinz_, Sep 22 2019