|
%I #52 Jun 27 2022 07:24:52
%S 1,1,2,7,34,206,1486,12412,117692,1248004,14625856,187638716,
%T 2614602112,39310384192,634148436104,10923398137576,200069534481616,
%U 3882002527006352,79535575126745632,1715658099715217584
%N Number of labeled forests of n nodes each component of which is a path.
%C 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
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (3.3.6).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(d).
%H T. D. Noe, <a href="/A011800/b011800.txt">Table of n, a(n) for n = 0..100</a>
%H Tobias Boege, Thomas Kahle, <a href="https://arxiv.org/abs/1902.11260">Construction Methods for Gaussoids</a>, arXiv:1902.11260 [math.CO], 2019.
%H Samuele Giraudo, <a href="https://arxiv.org/abs/1709.08416">Combalgebraic structures on decorated cliques</a>, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
%H J. Rasku, T. Karkkainen, P. Hotokka, <a href="https://www.researchgate.net/publication/256095631_Proceedings_of_the_Finnish_Operational_Research_Society_40th_Anniversary_Workshop">Solution Space Visualization as a Tool for Vehicle Routing Algorithm Development</a>, Proc. FORS-40, pp. 9-12, 2013.
%F E.g.f.: exp[ x + x^2/(2*(1 - x)) ].
%F a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A003724(k). - _Vladeta Jovovic_, Oct 19 2003
%F 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
%F a(n) ~ 2^(-3/4)*exp(sqrt(2*n)-n+1/4)*n^(n-1/4). - _Vaclav Kotesovec_, Oct 07 2012
%F 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
%t Function[ esl, esl*Array[ Factorial, Length[ esl ], 0 ] ][ CoefficientList[ Series[ Exp[ x+x^2/(2-2x) ], {x, 0, 20} ], x ] ] (* _Olivier Gérard_ *)
%t 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 *)
%o (Maxima)
%o 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 */
%K nonn,easy,nice
%O 0,3
%A Herbert S. Wilf
|
