A011800 Number of labeled forests of n nodes each component of which is a path.

1, 1, 2, 7, 34, 206, 1486, 12412, 117692, 1248004, 14625856, 187638716, 2614602112, 39310384192, 634148436104, 10923398137576, 200069534481616, 3882002527006352, 79535575126745632, 1715658099715217584

Offset

0,3

Comments

Denote the bivariate exponential g.f. by g(x,y)=exp(y*f(x)) where f(x)=(2x-x^2)/(2-2x). Then this sequence is the row sums of the array defined by the g.f. The differential dg/dy = f(x)*exp(y*f(x)) is the exponential generating function for an array with row sums in A201720. - R. J. Mathar, Jun 27 2022

References

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (3.3.6).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(d).

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Tobias Boege, Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.

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.

J. Rasku, T. Karkkainen, P. Hotokka, Solution Space Visualization as a Tool for Vehicle Routing Algorithm Development, Proc. FORS-40, pp. 9-12, 2013.

Formula

E.g.f.: exp[ x + x^2/(2*(1 - x)) ].

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A003724(k). - Vladeta Jovovic, Oct 19 2003

Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-1)^2*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012

a(n) ~ 2^(-3/4)*exp(sqrt(2*n)-n+1/4)*n^(n-1/4). - Vaclav Kotesovec, Oct 07 2012

a(n) = n!*Sum_{k=1..n} (Sum_{i=0..n-k} binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+k+i)*(-1)^(n-k-i))/k!, n > 0, a(0) = 1. - Vladimir Kruchinin, Nov 25 2012

Mathematica

Function[ esl, esl*Array[ Factorial, Length[ esl ], 0 ] ][ CoefficientList[ Series[ Exp[ x+x^2/(2-2x) ], {x, 0, 20} ], x ] ] (* Olivier Gérard *)
With[{nn=20}, CoefficientList[Series[Exp[x+x^2/(2*(1-x))], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, May 13 2019 *)

Prog

(Maxima)
a(n):=n!*sum(sum(binomial(k, n-k-i)*binomial(k+i-1, k-1)*2^(-n+k+i)*(-1)^(n-k-i), i, 0, n-k)/(k!), k, 1, n); /* Vladimir Kruchinin, Nov 25 2012 */

Crossrefs

Sequence in context: A056543 A357829 A075834 * A112916 A145845 A355292

Adjacent sequences: A011797 A011798 A011799 * A011801 A011802 A011803

Keyword

nonn,easy,nice

Author

Herbert S. Wilf

Status

approved