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%I #4 Jan 22 2019 07:45:41
%S 1,1,1,1,2,2,4,5,9,12,18,24,36,48,69,97,139,196,283,402,576,819,1161,
%T 1635,2301,3209,4469,6193,8571,11812,16291,22404,30850,42414,58393,
%U 80305,110578,152091,209308,287686,395352,542413,743603,1017489,1390510,1896482
%N Number of square multiset partitions of integer partitions of n.
%C A multiset partition is square if the number of parts is equal to the number of parts in each part.
%e The a(3) = 1 through a(9) = 12 square multiset partitions:
%e (3) (4) (5) (6) (7) (8) (9)
%e (11)(11) (21)(11) (21)(21) (22)(21) (22)(22) (32)(22)
%e (22)(11) (31)(21) (31)(22) (32)(31)
%e (31)(11) (32)(11) (31)(31) (33)(21)
%e (41)(11) (32)(21) (41)(22)
%e (33)(11) (41)(31)
%e (41)(21) (42)(21)
%e (42)(11) (43)(11)
%e (51)(11) (51)(21)
%e (52)(11)
%e (61)(11)
%e (111)(111)(111)
%t Table[Sum[Length[Union@@(Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@#]]&/@IntegerPartitions[n,{k}])],{k,Sqrt[n]}],{n,30}]
%Y Cf. A000219, A001970, A047968, A261049, A279787, A305551, A319066, A323580.
%K nonn
%O 0,5
%A _Gus Wiseman_, Jan 21 2019
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