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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).

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

Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.

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 *)

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

STATUS

approved

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Last modified January 21 23:23 EST 2019. Contains 319350 sequences. (Running on oeis4.)