cp's OEIS Frontend

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

%I A330472 #9 Jan 17 2023 18:22:22
%S A330472 1,0,1,0,4,2,0,10,8,3,0,33,48,18,5,0,91,204,118,32,7,0,298,959,743,
%T A330472 266,58,11,0,910,4193,4334,1927,519,94,15,0,3017,18947,25305,13992,
%U A330472 4407,966,154,22,0,9945,84798,145033,97947,36410,9023,1679,236,30
%N A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).
%H A330472 Andrew Howroyd, <a href="/A330472/b330472.txt">Table of n, a(n) for n = 0..350</a>
%e A330472 Triangle begins:
%e A330472    1
%e A330472    0   1
%e A330472    0   4   2
%e A330472    0  10   8   3
%e A330472    0  33  48  18   5
%e A330472    0  91 204 118  32   7
%e A330472    0 298 959 743 266  58  11
%e A330472 For example, row n = 3 counts the following multiset partitions:
%e A330472   {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
%e A330472   {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
%e A330472   {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
%e A330472   {{1}{11}}    {{2}}{{11}}
%e A330472   {{1}{12}}    {{1}}{{1}{1}}
%e A330472   {{1}{23}}    {{1}}{{1}{2}}
%e A330472   {{2}{11}}    {{1}}{{2}{3}}
%e A330472   {{1}{1}{1}}  {{2}}{{1}{1}}
%e A330472   {{1}{1}{2}}
%e A330472   {{1}{2}{3}}
%o A330472 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A330472 ColGf(k,n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A,k,x)*x^k + O(x*x^n), sExp(A)) ))}
%o A330472 M(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
%o A330472 { my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Jan 17 2023
%Y A330472 Row sums are A318566.
%Y A330472 Column k = 1 is A007716 (for n > 0).
%Y A330472 Column k = n is A000041.
%Y A330472 Partitions of partitions of partitions are A007713.
%Y A330472 Twice-factorizations are A050336.
%Y A330472 If this is the 3-dimensional version, the 2-dimensional version is A317533.
%Y A330472 See A330473 for a variation.
%Y A330472 Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.
%K A330472 nonn,tabl
%O A330472 0,5
%A A330472 _Gus Wiseman_, Dec 19 2019
%E A330472 Terms a(21) and beyond from _Andrew Howroyd_, Jan 17 2023