A358837 Number of odd-length multiset partitions of integer partitions of n.

0, 1, 2, 4, 7, 14, 28, 54, 106, 208, 399, 757, 1424, 2642, 4860, 8851, 15991, 28673, 51095, 90454, 159306, 279067, 486598, 844514, 1459625, 2512227, 4307409, 7357347, 12522304, 21238683, 35903463, 60497684, 101625958, 170202949, 284238857, 473356564, 786196353

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: ((1/Product_{k>=1} (1-x^k)^A000041(k)) - (1/Product_{k>=1} (1+x^k)^A000041(k))) / 2. - Andrew Howroyd, Dec 31 2022

Example

The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}    {{3}}          {{4}}            {{5}}
         {{1,1}}  {{1,2}}        {{1,3}}          {{1,4}}
                  {{1,1,1}}      {{2,2}}          {{2,3}}
                  {{1},{1},{1}}  {{1,1,2}}        {{1,1,3}}
                                 {{1,1,1,1}}      {{1,2,2}}
                                 {{1},{1},{2}}    {{1,1,1,2}}
                                 {{1},{1},{1,1}}  {{1,1,1,1,1}}
                                                  {{1},{1},{3}}
                                                  {{1},{2},{2}}
                                                  {{1},{1},{1,2}}
                                                  {{1},{2},{1,1}}
                                                  {{1},{1},{1,1,1}}
                                                  {{1},{1,1},{1,1}}
                                                  {{1},{1},{1},{1},{1}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@Reverse/@IntegerPartitions[n], OddQ[Length[#]]&]], {n, 0, 10}]

Prog

(PARI)
P(v, y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v, 1)) - Vec(P(v, -1)))/2} \\ Andrew Howroyd, Dec 31 2022

Crossrefs

The version for set partitions is A024429.

These multiset partitions are ranked by A026424.

The version for partitions is A027193.

The version for twice-partitions is A358824.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions, strict A296122.

Cf. A000219, A141199, A336342, A358334, A358831.

Sequence in context: A018330 A068060 A239791 * A251653 A057744 A251708

Adjacent sequences: A358834 A358835 A358836 * A358838 A358839 A358840

Keyword

nonn

Author

Gus Wiseman, Dec 05 2022

Extensions

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

Status

approved