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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.

%I A059441 #42 Mar 17 2021 17:45:43
%S A059441 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 A059441 105,3507,19355,19355,3507,105,1,1,0,30016,0,1024380,0,30016,0,1,1,
%U A059441 945,286884,11180820,66462606,66462606,11180820,286884,945,1
%N 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.
%D A059441 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
%H A059441 Andrew Howroyd, <a href="/A059441/b059441.txt">Table of n, a(n) for n = 1..300</a> (rows 1..24)
%H A059441 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 A059441 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 A059441 Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_graph">Regular graph</a>
%e A059441 1;
%e A059441 1,   1;
%e A059441 1,   0,       1;
%e A059441 1,   3,       3,        1;
%e A059441 1,   0,      12,        0,          1;
%e A059441 1,  15,      70,       70,         15,    1;
%e A059441 1,   0,     465,        0,        465,    0,   1;
%e A059441 1, 105,    3507,    19355,      19355, 3507, 105, 1;
%e A059441 1,   0,   30016,        0,    1024380, ...;
%e A059441 1, 945,  286884, 11180820,   66462606, ...;
%e A059441 1,   0, 3026655,        0, 5188453830, ...;
%t A059441 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 A059441 (PARI) for(n=1, 10, print(A059441(n))) \\ See A295193 for script, _Andrew Howroyd_, Aug 28 2019
%Y A059441 Row sums are A295193.
%Y A059441 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 A059441 Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
%Y A059441 Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.
%K A059441 tabl,nice,nonn
%O A059441 1,8
%A A059441 _N. J. A. Sloane_, Feb 01 2001
%E A059441 a(37)-a(55) from _Andrew Howroyd_, Aug 25 2017