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%I #4 Mar 14 2020 18:54:16
%S 1,2,3,12,118,9638,10622074,135037240786,18621890255342234,
%T 28688490385422625653266,511030957184968000138445253202
%N Number of n-regular graphs on 2n unlabeled vertices with half-edges.
%C A half-edge is like a loop except it only adds 1 to the degree of its vertex.
%C a(n) is the number of non-isomorphic 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero where isomorphism is up to simultaneous permutation of rows and columns. The case where rows and columns can be permuted independently is covered by A333165.
%F a(n) = A333161(2*n, n).
%F Cf. A333164, A333165.
%e The a(1) = 1 matrix is:
%e [+ -]
%e [- +]
%e .
%e The a(2) = 2 matrices are:
%e [+ + - -] [- - + +] [+ + - -]
%e [+ + - -] [- - + +] [+ - + -]
%e [- - + +] [+ + - -] [- + - +]
%e [- - + +] [+ + - -] [- - + +]
%Y Central coefficients of A333161.
%K nonn,more
%O 0,2
%A _Andrew Howroyd_, Mar 12 2020
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