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A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

%I A330473 #8 Jan 18 2023 12:19:31
%S A330473 1,0,1,0,2,4,0,3,8,10,0,5,28,38,33,0,7,56,146,152,91,0,11,138,474,786,
%T A330473 628,298,0,15,268,1388,3117,3808,2486,910,0,22,570,3843,11830,19147,
%U A330473 18395,9986,3017,0,30,1072,10094,40438,87081,110164,86388,39889,9945
%N A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.
%C A330473 As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.
%H A330473 Andrew Howroyd, <a href="/A330473/b330473.txt">Table of n, a(n) for n = 0..350</a>
%e A330473 Triangle begins:
%e A330473    1
%e A330473    0   1
%e A330473    0   2   4
%e A330473    0   3   8  10
%e A330473    0   5  28  38  33
%e A330473    0   7  56 146 152  91
%e A330473    0  11 138 474 786 628 298
%e A330473 For example, row n = 3 counts the following multiset partitions:
%e A330473   {{111}}  {{1}{11}}    {{1}{1}{1}}
%e A330473   {{112}}  {{1}{12}}    {{1}{1}{2}}
%e A330473   {{123}}  {{1}{23}}    {{1}{2}{3}}
%e A330473            {{2}{11}}    {{1}}{{1}{1}}
%e A330473            {{1}}{{11}}  {{1}}{{1}{2}}
%e A330473            {{1}}{{12}}  {{1}}{{2}{3}}
%e A330473            {{1}}{{23}}  {{2}}{{1}{1}}
%e A330473            {{2}}{{11}}  {{1}}{{1}}{{1}}
%e A330473                         {{1}}{{1}}{{2}}
%e A330473                         {{1}}{{2}}{{3}}
%o A330473 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A330473 ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}
%o A330473 M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
%o A330473 { my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Jan 18 2023
%Y A330473 Row sums are A318566.
%Y A330473 Column k = 1 is A000041 (for n > 0).
%Y A330473 Column k = n is A007716.
%Y A330473 Partitions of partitions of partitions are A007713.
%Y A330473 Twice-factorizations are A050336.
%Y A330473 The 2-dimensional version is A317533.
%Y A330473 See A330472 for a variation.
%Y A330473 Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.
%K A330473 nonn,tabl
%O A330473 0,5
%A A330473 _Gus Wiseman_, Dec 20 2019
%E A330473 Terms a(36) and beyond from _Andrew Howroyd_, Jan 18 2023