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 A060725 E.g.f.: exp(-(x^5/5))/(1-x). 7
 1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2975616, 32731776, 392781312, 5106157056, 71486198784, 1070549415936, 17128790654976, 291189441134592, 5241409940422656, 99586788868030464, 1991897970827821056, 41829857387384242176, 920256862522453327872 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 5-cycle. REFERENCES R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7. LINKS Harry J. Smith, Table of n, a(n) for n = 0..100 Plouffe, Simon, Exact formulas for integer sequences FORMULA 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. EXAMPLE 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. MAPLE for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 5^i)), i=0..floor(n/5))) od: PROG (PARI) { for (n=0, 100, write("b060725.txt", n, " ", n! * sum(i=0, n\5, (-1)^i / (i! * 5^i))); ) } [From 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 */ CROSSREFS Cf. A000142, A000090, A000138, A000266, A060725, A060726, A060727. Sequence in context: A053502 A053504 A215716 * A150299 A094012 A141253 Adjacent sequences:  A060722 A060723 A060724 * A060726 A060727 A060728 KEYWORD nonn AUTHOR Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001 EXTENSIONS More terms from James A. Sellers, Apr 24 2001 Entry improved by comments from Michael Somos, Jul 28 2009 STATUS approved

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