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A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.

%I A325834 #15 May 10 2021 07:41:05
%S A325834 0,0,1,1,3,3,6,7,12,14,21,21,37,43,51,56,90,98,130,143,180,200,230,
%T A325834 249,403,454,508,555,657,706,826,889,1295,1406,1568,1690,2194,2396,
%U A325834 2603,2841,3387,3672,4024,4344,4693,5079,5489,5840,9731,10424,11336,12093
%N A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.
%C A325834 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%C A325834 The Heinz numbers of these partitions are given by A325798.
%H A325834 Alois P. Heinz, <a href="/A325834/b325834.txt">Table of n, a(n) for n = 0..500</a>
%F A325834 a(n) = A000041(n) - A325831(n).
%F A325834 For n even, A325833(n) = a(n) - A325830(n/2); for n odd, A325833(n) = a(n).
%e A325834 The a(2) = 1 through a(9) = 14 partitions:
%e A325834   (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
%e A325834             (22)  (32)  (33)   (43)   (44)    (54)
%e A325834             (31)  (41)  (42)   (52)   (53)    (63)
%e A325834                         (51)   (61)   (62)    (72)
%e A325834                         (222)  (322)  (71)    (81)
%e A325834                         (411)  (331)  (332)   (333)
%e A325834                                (511)  (422)   (432)
%e A325834                                       (431)   (441)
%e A325834                                       (521)   (522)
%e A325834                                       (611)   (531)
%e A325834                                       (2222)  (621)
%e A325834                                       (5111)  (711)
%e A325834                                               (3222)
%e A325834                                               (6111)
%p A325834 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325834       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325834       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325834     end:
%p A325834 a:= n-> add(b(n$2, k), k=0..n):
%p A325834 seq(a(n), n=0..55);  # _Alois P. Heinz_, Aug 17 2019
%t A325834 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])<=n&]],{n,0,30}]
%t A325834 (* Second program: *)
%t A325834 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, Function[w, b[w, Min[w, i - 1], Quotient[p, j + 1]]][n - i*j], 0], {j, 0, n/i}]];
%t A325834 a[n_] := Sum[b[n, n, k], {k, 0, n}];
%t A325834 a /@ Range[0, 55] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A325834 Cf. A002033, A088880, A088881, A108917, A126796, A307699, A325694, A325792, A325798, A325828, A325830, A325831, A325832, A325833, A325836.
%K A325834 nonn
%O A325834 0,5
%A A325834 _Gus Wiseman_, May 29 2019