

A005225


Number of permutations of length n with equal cycles.
(Formerly M0903)


19



1, 2, 3, 10, 25, 176, 721, 6406, 42561, 436402, 3628801, 48073796, 479001601, 7116730336, 88966701825, 1474541093026, 20922789888001, 400160588853026, 6402373705728001, 133991603578884052, 2457732174030848001, 55735573291977790576, 1124000727777607680001
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OFFSET

1,2


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. P. Walsh, A differentiationbased characterization of primes, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #97511237.


LINKS



FORMULA

a(n) = n!*sum(((n/k)!*k^(n/k))^(1)) where sum is over all divisors k of n. Exponential generating function [for a(1) through a(n)]= sum(exp(t^k/k)1, k=1..n).
a(n) = (n1)! + 1 iff n is a prime.


EXAMPLE

For example, a(4)=10 since, of the 24 permutations of length 4, there are 6 permutations with consist of a single 4cycle, 3 permutations that consist of two 2cycles and 1 permutation with four 1cycles.
Also, a(7)=721 since there are 720 permutations with a single cycle of length 7 and 1 permutation with seven 1cycles.


MAPLE

a:= n> n!*add((d/n)^d/d!, d=numtheory[divisors](n)):


MATHEMATICA



PROG

(Maxima) a(n):= n!*lsum((d!*(n/d)^d)^(1), d, listify(divisors(n)));


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



