1,2

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

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.

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

T. D. Noe, Table of n, a(n) for n=1..50

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 37

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

E.g.f.: -ln(1+LambertW(-x)) - Vladeta Jovovic (vladeta(AT)Eunet.yu), 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

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.yu), Jul 21 2005)

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

a(n)=A000435(n) + n^(n-1). See also A063169.

Sequence in context: A025167 A136727 A120022 this_sequence A087885 A051442 A015735

Adjacent sequences: A001862 A001863 A001864 this_sequence A001866 A001867 A001868

nonn,easy,nice

njas

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 23 2000