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A060725
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E.g.f.: exp(-(x^5/5))/(1-x).
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7
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1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2975616, 32731776, 392781312, 5106157056, 71486198784, 1070549415936, 17128790654976, 291189441134592, 5241409940422656, 99586788868030464, 1991897970827821056, 41829857387384242176, 920256862522453327872
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 5-cycle.
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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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FORMULA
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The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/5 ]( (-1)^i /(i! * 5^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 5^i) = e^(-1/5) or a(n) ~ e^(-1/5) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/5) * (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=5, n>=0. - Simon Plouffe, Feb 18 2011
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EXAMPLE
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a(5) = 96 because in S_5 the permutations with no 5-cycle are the complement of the 24 5-cycles so a(5) = 5! - 24 = 96.
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MAPLE
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for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 5^i)), i=0..floor(n/5))) od:
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MATHEMATICA
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With[{nn=30}, CoefficientList[Series[Exp[-(x^5/5)]/(1-x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 24 2019 *)
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PROG
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(PARI) { for (n=0, 100, write("b060725.txt", n, " ", n! * sum(i=0, n\5, (-1)^i / (i! * 5^i))); ) } \\ Harry J. Smith, Jul 10 2009
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^5 / 5) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */
(PARI) { A060725_list(numterms) = Vec(serlaplace(exp(-x^5/5 + O(x^numterms))/(1-x))); } /* Eric M. Schmidt, Aug 22 2012 */
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CROSSREFS
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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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STATUS
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approved
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