A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1

Offset

1,8

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)

Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.

Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.

Wikipedia, Regular graph

Example

1;
1,   1;
1,   0,       1;
1,   3,       3,        1;
1,   0,      12,        0,          1;
1,  15,      70,       70,         15,    1;
1,   0,     465,        0,        465,    0,   1;
1, 105,    3507,    19355,      19355, 3507, 105, 1;
1,   0,   30016,        0,    1024380, ...;
1, 945,  286884, 11180820,   66462606, ...;
1,   0, 3026655,        0, 5188453830, ...;

Mathematica

Table[SeriesCoefficient[Product[1+Times@@x/@s, {s, Subsets[Range[n], {2}]}], Sequence@@Table[{x[i], 0, k}, {i, n}]], {n, 9}, {k, 0, n-1}] (* Gus Wiseman, Dec 24 2018 *)

Prog

(PARI) for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

Crossrefs

Row sums are A295193.

Columns: A123023 (k=1), A001205 (k=2), A002829 (k=3, with alternating zeros), A005815 (k=4), A338978 (k=5, with alternating zeros), A339847 (k=6).

Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).

Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.

Sequence in context: A283798 A065551 A283797 * A186028 A225054 A355604

Adjacent sequences: A059438 A059439 A059440 * A059442 A059443 A059444

Keyword

tabl,nice,nonn

Author

N. J. A. Sloane, Feb 01 2001

Extensions

a(37)-a(55) from Andrew Howroyd, Aug 25 2017

Status

approved