|
%I #42 Mar 17 2021 17:45:43
%S 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,
%T 105,3507,19355,19355,3507,105,1,1,0,30016,0,1024380,0,30016,0,1,1,
%U 945,286884,11180820,66462606,66462606,11180820,286884,945,1
%N 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
%H Andrew Howroyd, <a href="/A059441/b059441.txt">Table of n, a(n) for n = 1..300</a> (rows 1..24)
%H Denis S. Krotov, <a href="https://arxiv.org/abs/2012.00038">[[2,10],[6,6]]-equitable partitions of the 12-cube</a>, arXiv:2012.00038 [math.CO], 2020.
%H Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/LabelledEnumeration.pdf">Applications of a technique for labeled enumeration</a>, Congress. Numerantium, 40 (1983), 207-221. See page 216.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_graph">Regular graph</a>
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, 3, 3, 1;
%e 1, 0, 12, 0, 1;
%e 1, 15, 70, 70, 15, 1;
%e 1, 0, 465, 0, 465, 0, 1;
%e 1, 105, 3507, 19355, 19355, 3507, 105, 1;
%e 1, 0, 30016, 0, 1024380, ...;
%e 1, 945, 286884, 11180820, 66462606, ...;
%e 1, 0, 3026655, 0, 5188453830, ...;
%t 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 *)
%o (PARI) for(n=1, 10, print(A059441(n))) \\ See A295193 for script, _Andrew Howroyd_, Aug 28 2019
%Y Row sums are A295193.
%Y 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).
%Y Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
%Y Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.
%K tabl,nice,nonn
%O 1,8
%A _N. J. A. Sloane_, Feb 01 2001
%E a(37)-a(55) from _Andrew Howroyd_, Aug 25 2017
|
