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A262970 Total cycle length of all iteration trajectories of all elements of random mappings from [n] to [n]. 1
1, 10, 117, 1648, 27425, 528336, 11581885, 284878336, 7772592897, 233010784000, 7614411069221, 269412832512000, 10261487793254113, 418636033893726208, 18213563455467238125, 841799936112774086656, 41189866031118283907585, 2127207204243268173103104 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

An iteration trajectory is the directed graph obtained by iterating the mapping starting from one of the n elements until a cycle appears and consists of a tail attached to a cycle.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..380

P. Flajolet and A. M. Odlyzko, Random Mapping Statistics, INRIA RR 1114, 1989.

Math StackExchange, Generating functions for tail length and rho-length

FORMULA

E.g.f.: T/(1-T)^4, where T is the labeled tree function, average over all mappings and values asymptotic to sqrt(Pi*n/8).

MAPLE

proc(n) 1/2*n!*add(n^q*(n + 1 - q)*(n - q)/q!, q = 0 .. n - 1) end proc

MATHEMATICA

Table[n!/2 Sum[n^q (n + 1 - q) (n - q)/q!, {q, 0, n - 1}], {n, 21}] (* Michael De Vlieger, Oct 06 2015 *)

PROG

(PARI) a(n) = n! * sum(q=0, n-1, n^q*(n+1-q)*(n-q)/q!)/2;

CROSSREFS

Sequence in context: A251318 A083448 A024129 * A155622 A218501 A122887

Adjacent sequences:  A262967 A262968 A262969 * A262971 A262972 A262973

KEYWORD

nonn

AUTHOR

Marko Riedel, Oct 05 2015

STATUS

approved

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Last modified July 23 14:56 EDT 2017. Contains 289688 sequences.