%I #18 May 14 2021 08:20:05
%S 1,1,1,2,2,4,5,8,10,16,21,35,40,58,84,120,141,199,255,347,447,592,772,
%T 1006,1172,1504,1928,2455,3061,3859,4778,5953,7054,8737,10742,13193,
%U 15783,19241,23412,28344,33951,40911,49150,58917,70482,84055,100069,118914
%N Number of integer partitions of n whose number of submultisets is greater than n.
%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 A325795.
%H Alois P. Heinz, <a href="/A325831/b325831.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = A000041(n) - A325834(n).
%F For n even, a(n) = A325832(n) - A325830(n/2); for n odd, a(n) = A325832(n).
%e The a(1) = 1 through a(8) = 10 partitions:
%e (1) (11) (21) (211) (221) (321) (421) (3221)
%e (111) (1111) (311) (2211) (2221) (3311)
%e (2111) (3111) (3211) (4211)
%e (11111) (21111) (4111) (22211)
%e (111111) (22111) (32111)
%e (31111) (41111)
%e (211111) (221111)
%e (1111111) (311111)
%e (2111111)
%e (11111111)
%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-> combinat[numbpart](n)-add(b(n$2, k), k=0..n):
%p seq(a(n), n=0..55); # _Alois P. Heinz_, Aug 17 2019
%t Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])>n&]],{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,
%t Function[w, b[w, Min[w, i-1], p/(j+1)]][n-i*j], 0], {j, 0, n/i}]];
%t a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n}];
%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 13 2021, after _Alois P. Heinz_ *)
%Y Cf. A002033, A098859, A126796, A325694, A325792, A325795, A325828, A325830, A325832, A325833, A325834, A325836.
%K nonn
%O 0,4
%A _Gus Wiseman_, May 25 2019