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A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

%I A333330 #20 Mar 23 2020 21:12:20
%S A333330 1,1,1,1,0,1,1,0,1,1,1,0,1,0,1,1,0,1,1,1,1,1,0,1,0,2,0,1,1,0,1,1,3,2,
%T A333330 1,1,1,0,1,0,4,0,4,0,1,1,0,1,1,5,7,9,4,1,1,1,0,1,0,7,0,24,0,7,0,1,1,0,
%U A333330 1,1,8,16,54,60,32,8,1,1,1,0,1,0,10,0,128,0,240,0,12,0,1,1,0,1,1,12,37,271,955,1753,930,135,14,1,1
%N A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.
%C A333330 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 A333351. Burnside's lemma can be used to extend this method to the unlabeled case.
%H A333330 Andrew Howroyd, <a href="/A333330/b333330.txt">Table of n, a(n) for n = 0..350</a>
%e A333330 Array begins:
%e A333330 =================================================
%e A333330 n\k | 0 1 2  3   4    5      6     7        8
%e A333330 ----+--------------------------------------------
%e A333330   0 | 1 1 1  1   1    1      1     1        1 ...
%e A333330   1 | 1 0 0  0   0    0      0     0        0 ...
%e A333330   2 | 1 1 1  1   1    1      1     1        1 ...
%e A333330   3 | 1 0 1  0   1    0      1     0        1 ...
%e A333330   4 | 1 1 2  3   4    5      7     8       10 ...
%e A333330   5 | 1 0 2  0   7    0     16     0       37 ...
%e A333330   6 | 1 1 4  9  24   54    128   271      582 ...
%e A333330   7 | 1 0 4  0  60    0    955     0    12511 ...
%e A333330   8 | 1 1 7 32 240 1753  13467 90913   543779 ...
%e A333330   9 | 1 0 8  0 930    0 253373     0 35255015 ...
%e A333330   ...
%Y A333330 Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.
%Y A333330 Row n=4 is A001399.
%Y A333330 Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).
%K A333330 nonn,tabl
%O A333330 0,26
%A A333330 _Andrew Howroyd_, Mar 15 2020