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%I #8 Mar 12 2020 19:00:04
%S 1,1,1,1,2,1,1,2,2,1,1,3,3,3,1,1,3,4,4,3,1,1,4,8,12,8,4,1,1,4,10,24,
%T 24,10,4,1,1,5,17,70,118,70,17,5,1,1,5,24,172,634,634,172,24,5,1,1,6,
%U 36,525,4428,9638,4428,525,36,6,1,1,6,50,1530,35500,187990,187990,35500,1530,50,6,1
%N Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.
%C A half-edge is like a loop except it only adds 1 to the degree of its vertex.
%C T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.
%C T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.
%H Andrew Howroyd, <a href="/A333161/b333161.txt">Table of n, a(n) for n = 0..230</a>
%F T(n,k) = T(n, n-k).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 2, 1;
%e 1, 3, 3, 3, 1;
%e 1, 3, 4, 4, 3, 1;
%e 1, 4, 8, 12, 8, 4, 1;
%e 1, 4, 10, 24, 24, 10, 4, 1;
%e 1, 5, 17, 70, 118, 70, 17, 5, 1;
%e 1, 5, 24, 172, 634, 634, 172, 24, 5, 1;
%e 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
%e ...
%e The a(2,1) = 2 adjacency matrices are:
%e [0 1] [1 0]
%e [1 0] [0 1]
%e .
%e The A(4,2) = 3 adjacency matrices are:
%e [0 0 1 1] [1 1 0 0] [1 1 0 0]
%e [0 0 1 1] [1 1 0 0] [1 0 1 0]
%e [1 1 0 0] [0 0 1 1] [0 1 0 1]
%e [1 1 0 0] [0 0 1 1] [0 0 1 1]
%Y Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.
%Y Row sums are A333162.
%Y Central coefficients are A333166.
%Y Cf. A051031, A333157, A333159.
%K nonn,tabl
%O 0,5
%A _Andrew Howroyd_, Mar 11 2020
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