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%I #15 Sep 21 2022 23:27:35
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,5,3,1,1,1,5,8,10,3,1,1,1,6,14,26,
%T 16,4,1,1,1,7,20,61,60,29,4,1,1,1,8,30,128,243,184,45,5,1,1,1,9,40,
%U 254,800,1228,488,75,5,1,1,1,10,55,467,2518,7252,6684,1509,115,6,1
%N Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.
%C T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333737.
%C Terms may be computed without generating each graph by enumerating the graphs by degree sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A192517 can be used to extend this method to the unlabeled case.
%H Andrew Howroyd, <a href="/A333893/b333893.txt">Table of n, a(n) for n = 0..377</a> (antidiagonals 0..26)
%e Array begins:
%e ==============================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-----------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 2 3 4 5 6 7 8 ...
%e 3 | 1 2 5 8 14 20 30 40 ...
%e 4 | 1 3 10 26 61 128 254 467 ...
%e 5 | 1 3 16 60 243 800 2518 6999 ...
%e 6 | 1 4 29 184 1228 7252 38194 175369 ...
%e 7 | 1 4 45 488 6684 78063 772243 6254652 ...
%e ...
%Y Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
%Y Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
%Y Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.
%K nonn,tabl
%O 0,9
%A _Andrew Howroyd_, Apr 08 2020
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