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%I #8 Apr 09 2020 03:13:55
%S 1,1,5,12,33,74,163,319,596,1044,1754,2820,4394,6629,9756,14018,19743,
%T 27286,37111,49707,65699,85756,110691,141382,178882,224309,278990,
%U 344342,422000,513718,621506,747493,894105,1063920,1259823,1484886,1742533,2036377,2370431,2748936,3176544
%N Number of non-isomorphic 4 X 4 nonnegative integer symmetric matrices with all row and column sums equal to n up to permutations of rows and columns.
%e The a(2) = 5 matrices are:
%e [0 0 0 2] [0 0 0 2] [0 0 1 1] [0 0 1 1] [0 0 1 1]
%e [0 0 2 0] [0 1 1 0] [0 0 1 1] [0 1 0 1] [0 2 0 0]
%e [0 2 0 0] [0 1 1 0] [1 1 0 0] [1 0 1 0] [1 0 0 1]
%e [2 0 0 0] [2 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0]
%e .
%e The a(3) = 12 matrices are:
%e [0 0 0 3] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 1 1 1]
%e [0 0 3 0] [0 0 2 1] [0 1 1 1] [0 1 2 0] [0 2 1 0] [1 0 1 1]
%e [0 3 0 0] [1 2 0 0] [1 1 1 0] [1 2 0 0] [1 1 0 1] [1 1 0 1]
%e [3 0 0 0] [2 1 0 0] [2 1 0 0] [2 0 0 1] [2 0 1 0] [1 1 1 0]
%e .
%e [0 0 0 3] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 1 1 1] [1 0 1 1]
%e [0 1 2 0] [0 2 1 0] [0 3 0 0] [0 3 0 0] [1 1 0 1] [0 3 0 0]
%e [0 2 1 0] [1 1 1 0] [1 0 1 1] [1 0 2 0] [1 0 2 0] [1 0 1 1]
%e [3 0 0 0] [2 0 0 1] [2 0 1 0] [2 0 0 1] [1 1 0 1] [1 0 1 1]
%Y Row n=4 of A333737.
%K nonn
%O 0,3
%A _Andrew Howroyd_, Apr 08 2020