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A005225
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Number of permutations of length n with equal cycles.
(Formerly M0903)
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7
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1, 2, 3, 10, 25, 176, 721, 6406, 42561, 436402, 3628801, 48073796, 479001601, 7116730336, 88966701825, 1474541093026, 20922789888001, 400160588853026, 6402373705728001, 133991603578884052, 2457732174030848001
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n)=(n-1)!+1 iff n is a prime.
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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.
H. S. Wilf, Three problems in combinatorial asymptotics, J. Combin. Theory, A 35 (1983), 199-207.
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LINKS
| D. P. Walsh, Primality test based on the generating function
D. P. Walsh, A differentiation-based characterization of primes
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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), k=1..n).
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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.
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MATHEMATICA
| Table[n! Sum[((n/d)!*d^(n/d))^(-1), {d, Divisors[n]}], {n, 21}] (* From Jean-François Alcover, Apr 04 2011 *)
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CROSSREFS
| Sequence in context: A103018 A005158 A182926 * A052929 A151415 A134588
Adjacent sequences: A005222 A005223 A005224 * A005226 A005227 A005228
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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
| Additional comments from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Dec 08 2000
More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 01 2001
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