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%I #11 Apr 12 2020 15:18:17
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,3,5,5,1,1,1,1,3,9,12,7,
%T 1,1,1,1,4,13,33,29,11,1,1,1,1,4,20,74,142,79,15,1,1,1,1,5,28,163,556,
%U 742,225,22,1,1,1,1,5,39,319,1919,5369,4454,677,30,1,1
%N Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.
%C Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A318805 can be used to extend this method to the unlabeled case.
%H Andrew Howroyd, <a href="/A333737/b333737.txt">Table of n, a(n) for n = 0..377</a>
%e Array begins:
%e ==============================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-----------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 2 2 3 3 4 4 ...
%e 3 | 1 1 3 5 9 13 20 28 ...
%e 4 | 1 1 5 12 33 74 163 319 ...
%e 5 | 1 1 7 29 142 556 1919 5793 ...
%e 6 | 1 1 11 79 742 5369 31781 156191 ...
%e 7 | 1 1 15 225 4454 64000 692599 5882230 ...
%e ...
%e The T(3,3) = 5 matrices are:
%e [0 0 3] [0 1 2] [0 1 2] [1 0 2] [1 1 1]
%e [0 3 0] [1 1 1] [1 2 0] [0 3 0] [1 1 1]
%e [3 0 0] [2 1 0] [2 0 1] [2 0 1] [1 1 1]
%Y Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
%Y Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
%Y Main diagonal is A333738.
%Y Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
%Y Cf. A318805, A333893.
%K nonn,tabl
%O 0,13
%A _Andrew Howroyd_, Apr 08 2020
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