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

Author

N. J. A. Sloane, Feb 01 2001

Keywords

• tabl
• nice
• nonn

Examples

			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, ...;
		

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

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.

Programs

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 *)

PARI

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

Extensions

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