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Search: id:A060727
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| A060727 |
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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 7-cycle. |
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2
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| 1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 75613824000, 1134207360000, 18147317760000, 308504401920000, 5553079234560000, 105508505456640000, 2110170109132800000
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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This is the expansion of exp ((-x^7)/7)/(1-x).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,100
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FORMULA
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The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/7 ]( (-1)^i /(i! * 7^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 7^i) = e^(-1/7) or a(n) ~ e^(-1/7) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/7) * (n/e)^n * sqrt(2 * Pi * n)
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EXAMPLE
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a(7) = 4320 because in S_7 the permutations with no 7-cycle are the complement of the 720 7-cycles so a(7) = 7! - 720 = 4320.
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MAPLE
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for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 7^i)), i=0..floor(n/7))) od:
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PROGRAM
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(PARI) { for (n=0, 100, write("b060727.txt", n, " ", n! * sum(i=0, n\7, (-1)^i / (i! * 7^i))); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 10 2009]
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CROSSREFS
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A000142.
Sequence in context: A033644 A154654 A070947 this_sequence A152350 A152368 A152364
Adjacent sequences: A060724 A060725 A060726 this_sequence A060728 A060729 A060730
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KEYWORD
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nonn
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 24 2001
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