|
|
A129271
|
|
Number of labeled n-node connected graphs with at most one cycle.
|
|
45
|
|
|
1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.
Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024
|
|
REFERENCES
|
J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.
|
|
LINKS
|
|
|
FORMULA
|
a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016
|
|
EXAMPLE
|
a(4) = 16 + 3*3 = 31.
The a(0) = 1 through a(3) = 4 graph edge sets:
{} . {{1,2}} {{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
(End)
|
|
MAPLE
|
a := n -> `if`(n=0, 1, ((n-1)*exp(n)*GAMMA(n-1, n)+n^(n-2)*(3-n))/2):
|
|
MATHEMATICA
|
nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 23 2013 *)
|
|
PROG
|
(PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019
|
|
CROSSREFS
|
The non-connected non-covering version is A133686.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|