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A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

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