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A060726
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For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.
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6
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1, 1, 2, 6, 24, 120, 600, 4200, 33600, 302400, 3024000, 33264000, 405820800, 5275670400, 73859385600, 1107890784000, 17726252544000, 301346293248000, 5419293175296000, 102966570330624000, 2059331406612480000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This is the expansion of exp ((-x^6)/6) /(1-x).
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REFERENCES
| R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,100
Plouffe, Simon, Exact formulas for integer sequences
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FORMULA
| The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/6 ]( (-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (n/e)^n * sqrt(2 * Pi * n)
a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=6, n>=0. Simon Plouffe, Feb. 18 2011.
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EXAMPLE
| a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.
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MAPLE
| for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:
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PROG
| (PARI) { for (n=0, 100, write("b060726.txt", n, " ", n! * sum(i=0, n\6, (-1)^i / (i! * 6^i))); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 10 2009]
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CROSSREFS
| Cf. A000142 A000266 A000090 A000138 A060725 A060726 A060727
Sequence in context: A179357 A179364 A070946 * A152332 A152349 A152345
Adjacent sequences: A060723 A060724 A060725 * A060727 A060728 A060729
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KEYWORD
| nonn
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AUTHOR
| Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 24 2001
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