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

0,3

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.

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

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 *)

Crossrefs

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

Sequence in context: A093935 A099240 A134055 * A125698 A231495 A052447

Adjacent sequences: A136278 A136279 A136280 * A136282 A136283 A136284

Keyword

nonn

Author

Don Knuth, Mar 31 2008

Extensions

More terms from Vladeta Jovovic, May 20 2008

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

Status

approved