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A005225 Number of permutations of length n with equal cycles.
(Formerly M0903)
13
1, 2, 3, 10, 25, 176, 721, 6406, 42561, 436402, 3628801, 48073796, 479001601, 7116730336, 88966701825, 1474541093026, 20922789888001, 400160588853026, 6402373705728001, 133991603578884052, 2457732174030848001, 55735573291977790576, 1124000727777607680001 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = (n-1)! + 1 iff n is a prime.

REFERENCES

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

D. P. Walsh, A differentiation-based characterization of primes, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-237.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..200

D. P. Walsh, Primality test based on the generating function

D. P. Walsh, A differentiation-based characterization of primes

H. S. Wilf, Three problems in combinatorial asymptotics, J. Combin. Theory, A 35 (1983), 199-207.

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).

EXAMPLE

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

Also, a(7)=721 since there are 720 permutations with a single cycle of length 7 and 1 permutation with seven 1-cycles.

MAPLE

with(numtheory):

a:= n-> n!*add((d!*(n/d)^d)^(-1), d=divisors(n)):

seq(a(n), n=0..30);  # Alois P. Heinz, Nov 07 2012

MATHEMATICA

Table[n! Sum[((n/d)!*d^(n/d))^(-1), {d, Divisors[n]}], {n, 21}] (* Jean-Fran├žois Alcover, Apr 04 2011 *)

PROG

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

makelist(a(n), n, 1, 40); /* Emanuele Munarini, Feb 03 2014 */

CROSSREFS

Cf. A038041, A055225, A236696.

Column k=1 of A218868.

Sequence in context: A246437 A005158 A182926 * A211208 A238937 A278088

Adjacent sequences:  A005222 A005223 A005224 * A005226 A005227 A005228

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Dennis P. Walsh, Dec 08 2000

More terms from Vladeta Jovovic, Dec 01 2001

STATUS

approved

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Last modified May 28 02:51 EDT 2017. Contains 287211 sequences.