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

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, 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, 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

Offset

0,26

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..350

Example

Array begins:
=================================================
n\k | 0 1 2  3   4    5      6     7        8
----+--------------------------------------------
  0 | 1 1 1  1   1    1      1     1        1 ...
  1 | 1 0 0  0   0    0      0     0        0 ...
  2 | 1 1 1  1   1    1      1     1        1 ...
  3 | 1 0 1  0   1    0      1     0        1 ...
  4 | 1 1 2  3   4    5      7     8       10 ...
  5 | 1 0 2  0   7    0     16     0       37 ...
  6 | 1 1 4  9  24   54    128   271      582 ...
  7 | 1 0 4  0  60    0    955     0    12511 ...
  8 | 1 1 7 32 240 1753  13467 90913   543779 ...
  9 | 1 0 8  0 930    0 253373     0 35255015 ...
  ...

Crossrefs

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

Row n=4 is A001399.

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

Sequence in context: A326815 A117210 A060277 * A290825 A204688 A332038

Adjacent sequences: A333327 A333328 A333329 * A333331 A333332 A333333

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 15 2020

Status

approved