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%I #13 May 12 2021 21:03:17
%S 0,0,0,1,1,2,0,3,0,5,2,2,0,15,0,2,3,25,0,17,0,18,3,2,0,150,0,2,13,24,
%T 0,43,0,351,3,2,2,383,0,2,3,341,0,60,0,37,51,2,0,3733,0,31,3,42,0,460,
%U 1,633,3,2,0,1780,0,2,68,12460,0,87,0,55,3
%N Number of integer partitions of n having n - 1 different submultisets.
%C The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C The Heinz numbers of these partitions are given by A325694.
%H Alois P. Heinz, <a href="/A325836/b325836.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
%e (3) (22) (32) (322) (432) (3322) (32222) (4432)
%e (41) (331) (531) (4411) (71111) (5332)
%e (511) (621) (5422)
%e (3222) (5521)
%e (6111) (6322)
%e (6331)
%e (6511)
%e (7411)
%e (8221)
%e (8311)
%e (9211)
%e (33322)
%e (55111)
%e (322222)
%e (811111)
%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p end:
%p a:= n-> b(n$2,n-1):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Aug 17 2019
%t Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==n-1&]],{n,0,30}]
%t (* Second program: *)
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
%t If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
%t Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
%t a[n_] := b[n, n, n-1];
%t a /@ Range[0, 80] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)
%Y Positions of zeros are A307699.
%Y Cf. A002033, A088880, A088881, A108917, A325694, A325768, A325792, A325798, A325828, A325830, A325833, A325835.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 29 2019
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