

A263293


Triangle read by rows: T(n,k) is the number of graphs with n vertices and maximum vertex degree k, (0 <= k < n).


8



1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 1, 2, 8, 12, 11, 1, 3, 15, 43, 60, 34, 1, 3, 25, 121, 360, 378, 156, 1, 4, 41, 378, 2166, 4869, 3843, 1044, 1, 4, 65, 1095, 14306, 68774, 113622, 64455, 12346, 1, 5, 100, 3441, 104829, 1141597, 3953162, 4605833, 1921532, 274668
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OFFSET

1,6


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 A327366. Burnside's lemma can be used to extend this method to the unlabeled case.  Andrew Howroyd, Mar 10 2020


LINKS



FORMULA

G.f. for column k=0: A(x)=1/(1x).
G.f. for column k=1: B(x)=x^2/((1x^2)(1x)).
G.f. for column k=2: 1/((1x)(1x^2))*Product_{i>=3} 1/(1x^i)^2  B(x)  A(x).
(End)
T(n, 0) = 1.


EXAMPLE

Triangle begins:
1,
1, 1,
1, 1, 2,
1, 2, 4, 4,
1, 2, 8, 12, 11,
1, 3, 15, 43, 60, 34,
1, 3, 25, 121, 360, 378, 156,
1, 4, 41, 378, 2166, 4869, 3843, 1044,
...


CROSSREFS

Row sums are A000088 (simple graphs on n nodes).
Cf. A294217 (triangle of nnode minimum vertex degree counts).


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



