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Number of multisets of nonempty sets of nonempty sets of positive integers with total sum n.
1

%I #4 Dec 18 2019 09:01:48

%S 1,1,2,5,10,20,43,84,168,332,650,1255,2428,4636,8827,16702,31457,

%T 58919,109977,204286,378135,697240,1281315,2346612,4284654,7799248,

%U 14157079,25626996,46269838,83330373,149717844,268371413,479996794,856661792,1525761119,2712050472

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

%F Euler transform of A050342. 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.

%e The a(4) = 10 partitions:

%e ((4)) ((13)) ((1)(12)) ((2))((2)) ((1))((1))((1))((1))

%e ((1)(3)) ((1))((12))

%e ((1))((3)) ((1))((1)(2))

%e ((1))((1))((2))

%t ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];

%t Table[Length[Select[ppl[n,3],And[And@@UnsameQ@@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

%Y Cf. A001055, A001970, A007713, A050342, A050343, A063834, A089259, A270995, A279785, A330461, A323787-A323795, A330452-A330459.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 17 2019