A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890

Offset

0,7

Links

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

Formula

Euler transform of A083751. - Andrew Howroyd, Oct 25 2018

Example

The a(4) = 1 through a(10) = 15 multiset partitions:
  ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
                  ((33))   ((34))   ((35))      ((36))      ((37))
                  ((222))  ((223))  ((44))      ((45))      ((46))
                                    ((224))     ((225))     ((55))
                                    ((233))     ((234))     ((226))
                                    ((2222))    ((333))     ((235))
                                    ((22)(22))  ((2223))    ((244))
                                                ((22)(23))  ((334))
                                                            ((2224))
                                                            ((2233))
                                                            ((22222))
                                                            ((22)(24))
                                                            ((22)(33))
                                                            ((23)(23))
                                                            ((22)(222))

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/@Select[IntegerPartitions[n], FreeQ[#, 1]&], FreeQ[Length/@#, 1]&]], {n, 20}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018

Crossrefs

Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

Sequence in context: A086543 A281616 A110618 * A320294 A304179 A182559

Adjacent sequences: A320288 A320289 A320290 * A320292 A320293 A320294

Keyword

nonn

Author

Gus Wiseman, Oct 09 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved