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%I #17 Mar 23 2020 15:44:45
%S 1,1,0,1,1,1,1,0,2,0,1,1,3,2,1,1,0,5,0,3,0,1,1,7,8,7,3,1,1,0,11,0,20,
%T 0,4,0,1,1,15,31,56,32,13,4,1,1,0,22,0,187,0,66,0,5,0,1,1,30,140,654,
%U 727,384,101,22,5,1,1,0,42,0,2705,0,3369,0,181,0,6,0,1,1,56,722,12587,42703
%N Square array T(n,k), read by upward antidiagonals, counting isomorphism classes of k-regular multigraphs of order n, loops allowed.
%C The number of vertices n is positive; valency k is nonnegative.
%C Each loop contributes two to the valency of its vertex.
%C The antidiagonal having coordinate sum t=n+k is read from T(t,0) to T(1,t-1).
%C Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333467. Burnside's lemma can be used to extend this method to the unlabeled case. - _Andrew Howroyd_, Mar 23 2020
%H Andrew Howroyd, <a href="/A167625/b167625.txt">Table of n, a(n) for n = 1..378</a> (27 antidiagonals, first 19 antidiagonals from Jason Kimberley)
%H J. S. Kimberley, <a href="http://oeis.org/wiki/User:Jason_Kimberley/A167625">Table in user subpage of wiki</a>.
%H R. C. Read, <a href="http://dx.doi.org/10.1112/jlms/s1-34.4.417">The enumeration of locally restricted graphs (I)</a>, J. London Math. Soc. 34 (1959) 417-436.
%F T(n,k) = N\{S_n[S_k] * S_{nk/2}[S_2]\}.
%e Array begins:
%e ==============================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-----------------------------------------
%e 1 | 1 0 1 0 1 0 1 0 ...
%e 2 | 1 1 2 2 3 3 4 4 ...
%e 3 | 1 0 3 0 7 0 13 0 ...
%e 4 | 1 1 5 8 20 32 66 101 ...
%e 5 | 1 0 7 0 56 0 384 0 ...
%e 6 | 1 1 11 31 187 727 3369 12782 ...
%e 7 | 1 0 15 0 654 0 40365 0 ...
%e 8 | 1 1 22 140 2705 42703 675368 8584767 ...
%e ...
%Y Column sequences: A000012 (k=0), A059841 (k=1), A000041 (k=2), A129427 (k=3), A129429 (k=4), A129431 (k=5), A129433 (k=6), A129435 (k=7), A129437 (k=8).
%Y Cf. A333330 (loopless), A333397 (connected), A333467 (labeled).
%K nonn,tabl
%O 1,9
%A _Jason Kimberley_, Nov 07 2009
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