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%I #9 Jan 17 2023 12:36:52
%S 1,0,2,3,19,39,200,615,2849,11174,52377,239269,1191090,6041975,
%T 32275288,177797719,1017833092,6014562272,36717301665,230947360981,
%U 1495562098099,9956230757240,68070158777759,477439197541792,3432259679880648,25267209686664449
%N Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.
%C All sets and multisets must be finite, and only the outermost may be empty.
%C The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.
%e Non-isomorphic representatives of the a(4) = 19 multiset partitions:
%e {{1111}} {{1112}} {{1123}} {{1234}}
%e {{11}{11}} {{1122}} {{11}{23}} {{12}{34}}
%e {{11}}{{11}} {{11}{12}} {{12}{13}} {{12}}{{34}}
%e {{11}{22}} {{11}}{{23}}
%e {{12}{12}} {{12}}{{13}}
%e {{11}}{{12}}
%e {{11}}{{22}}
%e {{12}}{{12}}
%e Non-isomorphic representatives of the a(5) = 39 multiset partitions:
%e {{11111}} {{11112}} {{11123}} {{11234}} {{12345}}
%e {{11}{111}} {{11122}} {{11223}} {{11}{234}} {{12}{345}}
%e {{11}}{{111}} {{11}{112}} {{11}{123}} {{12}{134}} {{12}}{{345}}
%e {{11}{122}} {{11}{223}} {{23}{114}}
%e {{12}{111}} {{12}{113}} {{11}}{{234}}
%e {{12}{112}} {{12}{123}} {{12}}{{134}}
%e {{22}{111}} {{13}{122}} {{23}}{{114}}
%e {{11}}{{112}} {{23}{111}}
%e {{11}}{{122}} {{11}}{{123}}
%e {{12}}{{111}} {{11}}{{223}}
%e {{12}}{{112}} {{12}}{{113}}
%e {{22}}{{111}} {{12}}{{123}}
%e {{13}}{{122}}
%e {{23}}{{111}}
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp(sExp(A-x*sv(1)))))} \\ _Andrew Howroyd_, Jan 17 2023
%Y Cf. A007716, A302545, A306186, A317791, A318564, A318566.
%Y Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 28 2019
%E Terms a(8) and beyond from _Andrew Howroyd_, Jan 17 2023
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