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 A001865 Number of connected functions on n labeled nodes. (Formerly M3040 N1232) 17

%I M3040 N1232

%S 1,3,17,142,1569,21576,355081,6805296,148869153,3660215680,

%T 99920609601,2998836525312,98139640241473,3478081490967552,

%U 132705415800984825,5423640496274200576,236389784118231290049,10944997108429625524224

%N Number of connected functions on n labeled nodes.

%C If one randomly selects a ball from an urn containing n different balls, with replacement, until exactly one ball has been selected twice, the probability that that ball was also the first ball selected once is a(n)/n^n. See also A000435. - _Matthew Vandermast_, Jun 15 2004

%C a(n) equals the permanent of the (n-1)X(n-1) matrix with n+1's along the main diagonal and 1's everywhere else. - _John M. Campbell_, Apr 20 2012

%D L. Katz, Probability of indecomposability of a random mapping function. Ann. Math. Statist. 26, (1955), 512-517.

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 112.

%D Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994

%D F. Schmidt and R. Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D BERND STURMFELS AND NGOC MAI TRAN, COMBINATORIAL TYPES OF TROPICAL EIGENVECTORS, Arxiv preprint arXiv:1105.5504, 2011.

%H T. D. Noe, <a href="/A001865/b001865.txt">Table of n, a(n) for n=1..50</a>

%H Christian Brouder, William J. Keith, Ângela Mestre, <a href="http://arxiv.org/abs/1301.0874">Closed forms for a multigraph enumeration</a>, arXiv preprint arXiv:1301.0874, 2013.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=37">Encyclopedia of Combinatorial Structures 37</a>

%F Sum n! n^(n-k-1) / (n-k)!, k = 1 . . n.

%F E.g.f.: -ln(1+LambertW(-x)). - _Vladeta Jovovic_, Apr 11 2001

%F E.g.f. satisfies 0=2y'^4+2y''^2-y'''y'-y''y'^2. - _Michael Somos_, Aug 23 2003

%F Integral representation in terms of incomplete Gamma function: a(n)=exp(n+1)*Integral_{x=n+1..infinity} x^n exp(-x) dx.

%F Asymptotics: sqrt(Pi*n/2)*n^(n-1) - _N-E. Fahssi_, Jan 25 2008, corrected by _Vaclav Kotesovec_, Nov 27 2012

%F a(n) = exp(1)*Integral_{x=1..infinity} (n+x)^n*exp(-x) dx. - _Gerald McGarvey_, Apr 16 2008

%p spec := [B, {A=Prod(Z,Set(A)), B=Cycle(A)}, labeled]; [seq(combstruct[count](spec,size=n), n=0..20)];

%p seq(simplify(GAMMA(n,n)*exp(n)),n=1..20); # _Vladeta Jovovic_, Jul 21 2005

%t t=Sum[n^(n-1)x^n/n!,{n,1,20}];

%t Range[0,20]! CoefficientList[Series[Log[1/(1-t)]+1,{x,0,20}],x] (* _Geoffrey Critzer_, Mar 12 2011 *)

%t f[n_] := Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]; Array[f, 18] (* _Robert G. Wilson v_ *)

%t a[n_] := Exp[n]*Gamma[n, n]; Table[a[n] // FunctionExpand, {n, 1, 18}] (* _Jean-François Alcover_, May 13 2013, after _Vladeta Jovovic_ *)

%o (PARI) a(n)=if(n<0,0,n!*sum(k=1,n,n^(n-k-1)/(n-k)!))

%K nonn,easy,nice,changed

%O 1,2

%A _N. J. A. Sloane_.

%E More terms from _James A. Sellers_, May 23 2000

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