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%I #16 May 11 2021 06:14:25
%S 1,1,1,2,1,3,1,4,2,3,1,12,1,3,4,21,1,14,1,18,4,3,1,116,3,3,12,25,1,40,
%T 1,271,4,3,4,325,1,3,4,295,1,56,1,36,47,3,1,3128,4,32,4,44,1,407,4,
%U 566,4,3,1,1598,1,3,65,10656,5,90,1,54,4,84,1
%N Number of integer partitions of n having exactly n + 1 submultisets.
%C The Heinz numbers of these partitions are given by A325792.
%C The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
%H Alois P. Heinz, <a href="/A325828/b325828.txt">Table of n, a(n) for n = 0..1000</a>
%e The 12 = 11 + 1 submultisets of the partition (4331) are: (), (1), (3), (4), (31), (33), (41), (43), (331), (431), (433), (4331), so (4331) is counted under a(11).
%e The a(5) = 3 through a(11) = 12 partitions:
%e 221 111111 421 3311 22221 1111111111 4322
%e 311 2221 11111111 51111 4331
%e 11111 4111 111111111 4421
%e 1111111 5411
%e 6221
%e 6311
%e 7211
%e 33311
%e 44111
%e 222221
%e 611111
%e 11111111111
%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[#])-1==n&]],{n,0,30}]
%t (* Second program: *)
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = Quotient[p, j + 1]; 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 11 2021, after _Alois P. Heinz_ *)
%Y Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, A325833, A325834, A325836.
%K nonn
%O 0,4
%A _Gus Wiseman_, May 25 2019
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