A136281 Number of graphs on n labeled nodes with degree at most 2.

1, 1, 2, 8, 41, 253, 1858, 15796, 152219, 1638323, 19467494, 252998224, 3568259503, 54263159347, 884834059454, 15397757661092, 284767413357977, 5576696746139689, 115269732256964626, 2507575465491619672, 57262481225957071721, 1369461739453440893261

Offset: 0

Author

Don Knuth, Mar 31 2008

Keywords

• nonn

Comments

These are thunderstorm graphs. Their connected components are a single cycle (clouds), a path (lightning bolts) or an isolated vertex (raindrops). - Geoffrey Critzer, May 11 2011

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..445 (terms n=1..200 from Vincenzo Librandi)
• Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.

Crossrefs

Cf. A000085 (degree at most 1), A136282-A136286.

Programs

Mathematica

f = (Log[1/(1-x)]+1/(1-x) -x^2/2 - 1)/2;
Range[0,25]! CoefficientList[Series[Exp[f],{x,0,25}],x] (* Geoffrey Critzer, May 11 2011 *)

Formula

Binomial transform of A000986. E.g.f.: (1-x)^(-1/2)*exp(-x^2/4 + x/((2*(1-x)))). - Vladeta Jovovic, May 20 2008

a(n) = (2*n-1)*a(n-1) - (n-1)^2*a(n-2) + (n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)/2*a(n-4). - Vaclav Kotesovec, Aug 10 2013

a(n) ~ n^n*exp(sqrt(2*n)-1/2-n)/sqrt(2) * (1+19/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 10 2013

Extensions

More terms from Vladeta Jovovic, May 20 2008

a(0)=1 prepended by Alois P. Heinz, Jul 21 2021