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A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

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