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 A325828 Number of integer partitions of n having exactly n + 1 submultisets. 12
 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, 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, 566, 4, 3, 1, 1598, 1, 3, 65, 10656, 5, 90, 1, 54, 4, 84, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The Heinz numbers of these partitions are given by A325792. The number of submultisets of an integer partition is the product of its multiplicities, each plus one. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 EXAMPLE 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). The a(5) = 3 through a(11) = 12 partitions:   221    111111  421      3311      22221      1111111111  4322   311            2221     11111111  51111                  4331   11111          4111               111111111              4421                  1111111                                   5411                                                            6221                                                            6311                                                            7211                                                            33311                                                            44111                                                            222221                                                            611111                                                            11111111111 MAPLE b:= proc(n, i, p) option remember; `if`(n=0 or i=1,       `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,       (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))     end: a:= n-> b(n\$2, n+1): seq(a(n), n=0..80);  # Alois P. Heinz, Aug 17 2019 MATHEMATICA Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])-1==n&]], {n, 0, 30}] (* Second program: *) 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}]]; a[n_] := b[n, n, n+1]; a /@ Range[0, 80] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *) CROSSREFS Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, A325831, A325832, A325833, A325834, A325836. Sequence in context: A173801 A108466 A211110 * A200780 A338899 A194943 Adjacent sequences:  A325825 A325826 A325827 * A325829 A325830 A325831 KEYWORD nonn AUTHOR Gus Wiseman, May 25 2019 STATUS approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)