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Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.
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%I #20 Mar 23 2020 21:12:20

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

%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 A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

%H Andrew Howroyd, <a href="/A333330/b333330.txt">Table of n, a(n) for n = 0..350</a>

%e Array begins:

%e =================================================

%e n\k | 0 1 2 3 4 5 6 7 8

%e ----+--------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 1 ...

%e 1 | 1 0 0 0 0 0 0 0 0 ...

%e 2 | 1 1 1 1 1 1 1 1 1 ...

%e 3 | 1 0 1 0 1 0 1 0 1 ...

%e 4 | 1 1 2 3 4 5 7 8 10 ...

%e 5 | 1 0 2 0 7 0 16 0 37 ...

%e 6 | 1 1 4 9 24 54 128 271 582 ...

%e 7 | 1 0 4 0 60 0 955 0 12511 ...

%e 8 | 1 1 7 32 240 1753 13467 90913 543779 ...

%e 9 | 1 0 8 0 930 0 253373 0 35255015 ...

%e ...

%Y Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.

%Y Row n=4 is A001399.

%Y Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

%K nonn,tabl

%O 0,26

%A _Andrew Howroyd_, Mar 15 2020