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A001865
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Number of connected functions on n labeled nodes.
(Formerly M3040 N1232)
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13
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1, 3, 17, 142, 1569, 21576, 355081, 6805296, 148869153, 3660215680, 99920609601, 2998836525312, 98139640241473, 3478081490967552, 132705415800984825, 5423640496274200576, 236389784118231290049, 10944997108429625524224
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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 (ghodges14(AT)comcast.net), Jun 15 2004
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REFERENCES
| L. Katz, Probability of indecomposability of a random mapping function. Ann. Math. Statist. 26, (1955), 512-517.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 112.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
F. Schmidt and R. Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
BERND STURMFELS AND NGOC MAI TRAN, COMBINATORIAL TYPES OF TROPICAL EIGENVECTORS, Arxiv preprint arXiv:1105.5504, 2011.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..50
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 37
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FORMULA
| Sum n! n^(n-k-1) / (n-k)!, k = 1 . . n.
E.g.f.: -ln(1+LambertW(-x)) - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 11 2001
E.g.f. satisfies 0=2y'^4+2y''^2-y'''y'-y''y'^2. - Michael Somos, Aug 23 2003
Integral representation in terms of incomplete Gamma function : a(n)=exp(n+1)*Integral_{x=n+1..\infty} x^n exp(-x) dx ; Asymptotics : exp(1)*sqrt(Pi*n/2)*n^n - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Jan 25 2008
a(n) = exp(1)*Integral_{x=1..\infty} (n+x)^n*exp(-x) dx - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 16 2008
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MAPLE
| spec := [B, {A=Prod(Z, Set(A)), B=Cycle(A)}, labeled]; [seq(combstruct[count](spec, size=n), n=0..20)];
seq(simplify(GAMMA(n, n)*exp(n)), n=1..20); (Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 21 2005)
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MATHEMATICA
| t=Sum[n^(n-1)x^n/n!, {n, 1, 20}];
Range[0, 20]! CoefficientList[Series[Log[1/(1-t)]+1, {x, 0, 20}], x] (* Geoffrey Critzer, Mar 12 2011 *)
f[n_] := Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]; Array[f, 18] (* RGWv *)
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PROG
| (PARI) a(n)=if(n<0, 0, n!*sum(k=1, n, n^(n-k-1)/(n-k)!))
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CROSSREFS
| a(n)=A000435(n) + n^(n-1). See also A063169.
Sequence in context: A136727 A062873 A120022 * A189001 A087885 A178685
Adjacent sequences: A001862 A001863 A001864 * A001866 A001867 A001868
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 23 2000
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