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A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

%I A333893 #15 Sep 21 2022 23:27:35
%S A333893 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 A333893 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 A333893 254,800,1228,488,75,5,1,1,1,10,55,467,2518,7252,6684,1509,115,6,1
%N A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.
%C A333893 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 A333893 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 A333893 Andrew Howroyd, <a href="/A333893/b333893.txt">Table of n, a(n) for n = 0..377</a> (antidiagonals 0..26)
%e A333893 Array begins:
%e A333893 ==============================================
%e A333893 n\k | 0 1  2   3    4     5      6       7
%e A333893 ----+-----------------------------------------
%e A333893   0 | 1 1  1   1    1     1      1       1 ...
%e A333893   1 | 1 1  1   1    1     1      1       1 ...
%e A333893   2 | 1 2  3   4    5     6      7       8 ...
%e A333893   3 | 1 2  5   8   14    20     30      40 ...
%e A333893   4 | 1 3 10  26   61   128    254     467 ...
%e A333893   5 | 1 3 16  60  243   800   2518    6999 ...
%e A333893   6 | 1 4 29 184 1228  7252  38194  175369 ...
%e A333893   7 | 1 4 45 488 6684 78063 772243 6254652 ...
%e A333893   ...
%Y A333893 Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
%Y A333893 Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
%Y A333893 Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.
%K A333893 nonn,tabl
%O A333893 0,9
%A A333893 _Andrew Howroyd_, Apr 08 2020