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%I #19 Jan 16 2024 19:52:24
%S 1,1,2,4,9,17,36,72,155,319,677,1429,3094,6648,14518,31796,70491,
%T 156818,352371,795952,1813580,4155367,9594425,22283566,52122379,
%U 122631874,290432439,691831161,1658270316,3997272089,9692519896,23631827354,57943821449,142834652193
%N Number of non-isomorphic self-dual multiset partitions of weight n.
%C Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.
%C The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.
%H Andrew Howroyd, <a href="/A316983/b316983.txt">Table of n, a(n) for n = 0..50</a>
%e Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
%e (1111),
%e (1)(222), (2)(122), (11)(22), (12)(12),
%e (1)(1)(23), (1)(2)(33), (1)(3)(23),
%e (1)(2)(3)(4).
%e The a(4) = 9 square symmetric matrices:
%e . [4]
%e .
%e . [3 0] [2 0] [2 1] [1 1]
%e . [0 1] [0 2] [1 0] [1 1]
%e .
%e . [2 0 0] [1 1 0] [0 1 1]
%e . [0 1 0] [1 0 0] [1 0 0]
%e . [0 0 1] [0 0 1] [1 0 0]
%e .
%e . [1 0 0 0]
%e . [0 1 0 0]
%e . [0 0 1 0]
%e . [0 0 0 1]
%o (PARI) vector(25, n, n--; T(n,n)) \\ T(n,k) defined in A318805. - _Andrew Howroyd_, Jan 16 2024
%Y Row sums of A320796.
%Y Main diagonal of A318805.
%Y Cf. A000009, A001055, A007716, A007717, A020555, A045778.
%Y Cf. A316974, A316978, A316979, A316980, A316981.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 18 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Sep 03 2018
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