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A011800 Number of labeled forests of n nodes each component of which is a path. 4
1, 1, 2, 7, 34, 206, 1486, 12412, 117692, 1248004, 14625856, 187638716, 2614602112, 39310384192, 634148436104, 10923398137576, 200069534481616, 3882002527006352, 79535575126745632, 1715658099715217584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

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

J. Rasku, T. Karkkainen, P. Hotokka, Solution Space Visualization as a Tool for Vehicle Routing Algorithm Development, Proc. FORS-40, 2013; http://www.fors40.org/wp-content/uploads/2013/01/FORS40_Proceedings_ISBN_978-952-265-436-6.pdf#page=14

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

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 *)

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: A212027 A056543 A075834 * A112916 A145845 A002720

Adjacent sequences:  A011797 A011798 A011799 * A011801 A011802 A011803

KEYWORD

nonn,easy,nice

AUTHOR

Herbert S. Wilf (wilf(AT)math.upenn.edu)

STATUS

approved

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Last modified March 27 16:36 EDT 2017. Contains 284177 sequences.