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%I #15 Jan 16 2024 19:52:41
%S 1,1,1,1,2,1,1,4,3,1,1,5,7,3,1,1,7,14,10,3,1,1,9,23,24,11,3,1,1,12,39,
%T 53,34,12,3,1,1,14,61,102,86,39,12,3,1,1,17,90,193,201,117,42,12,3,1,
%U 1,20,129,340,434,310,136,43,12,3,1,1,24,184,584,902,778,412,149,44,12,3,1
%N Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.
%C Also the number of nonnegative integer k X k symmetric matrices with sum of elements equal to n and no zero rows or columns, up to row and column permutations.
%C The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H Andrew Howroyd, <a href="/A320796/b320796.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F T(n,k) = A318805(k,n) - A318805(k-1,n). - _Andrew Howroyd_, Jan 16 2024
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 4 3 1
%e 1 5 7 3 1
%e 1 7 14 10 3 1
%e 1 9 23 24 11 3 1
%e 1 12 39 53 34 12 3 1
%e 1 14 61 102 86 39 12 3 1
%e 1 17 90 193 201 117 42 12 3 1
%e Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):
%e 1: {{1}}
%e .
%e 2: {{11}} {{1}{2}}
%e .
%e 3: {{111}} {{1}{22}} {{1}{2}{3}}
%e . {{2}{12}}
%e .
%e 4: {{1111}} {{11}{22}} {{1}{1}{23}} {{1}{2}{3}{4}}
%e . {{12}{12}} {{1}{2}{33}}
%e . {{1}{222}} {{1}{3}{23}}
%e . {{2}{122}}
%e .
%e 5: {{11111}} {{11}{122}} {{1}{22}{33}} {{1}{2}{2}{34}} {{1}{2}{3}{4}{5}}
%e . {{11}{222}} {{1}{23}{23}} {{1}{2}{3}{44}}
%e . {{12}{122}} {{1}{2}{333}} {{1}{2}{4}{34}}
%e . {{1}{2222}} {{1}{3}{233}}
%e . {{2}{1222}} {{2}{12}{33}}
%e . {{2}{13}{23}}
%e . {{3}{3}{123}}
%o (PARI) row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - _Andrew Howroyd_, Jan 16 2024
%Y Row sums are A316983.
%Y Cf. A000219, A007716, A316980, A317533, A318805, A319560, A319616, A319721, A320797-A320813.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, Nov 02 2018
%E a(56) onwards from _Andrew Howroyd_, Jan 16 2024
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