A330458 Number of multisets of nonempty sets of nonempty multisets of positive integers with total sum n.

1, 1, 3, 8, 20, 49, 123, 292, 701, 1653, 3874, 8977, 20711, 47344, 107692, 243382, 547264, 1224048, 2725483, 6040796, 13334354, 29316445, 64215841, 140159357, 304890958, 661097630, 1429083295, 3080159882, 6620188725, 14190463947, 30338920339, 64702805452

Offset: 0

Author

Gus Wiseman, Dec 17 2019

Keywords

• nonn

Examples

			The a(4) = 20 partitions:
  ((4))  ((22))      ((13))      ((112))          ((1111))
         ((2))((2))  ((1)(3))    ((1)(12))        ((1)(111))
                     ((1))((3))  ((2)(11))        ((1))((111))
                                 ((1))((12))      ((11))((11))
                                 ((2))((11))      ((1))((1)(11))
                                 ((1))((1)(2))    ((1))((1))((11))
                                 ((1))((1))((2))  ((1))((1))((1))((1))
		

Crossrefs

Cf. A001970, A007713, A050342, A050343, A063834, A089259, A261049, A270995, A271619, A323787-A323795, A330452-A330459.

Programs

Mathematica

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And@@UnsameQ@@@#&]],{n,0,10}]

Formula

Euler transform of A261049. The Euler transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} 1/(1 - x^i)^s_i.