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%I #27 Aug 24 2019 15:58:24
%S 1,1,2,6,24,96,576,4032,32256,290304,2975616,32731776,392781312,
%T 5106157056,71486198784,1070549415936,17128790654976,291189441134592,
%U 5241409940422656,99586788868030464,1991897970827821056,41829857387384242176,920256862522453327872
%N E.g.f.: exp(-(x^5/5))/(1-x).
%C a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 5-cycle.
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
%H Harry J. Smith, <a href="/A060725/b060725.txt">Table of n, a(n) for n = 0..100</a>
%H Plouffe, Simon, <a href="http://www.plouffe.fr/simon/exact.htm">Exact formulas for integer sequences</a>
%F 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).
%F 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
%e 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.
%p for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 5^i)), i=0..floor(n/5))) od:
%t With[{nn=30},CoefficientList[Series[Exp[-(x^5/5)]/(1-x),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Aug 24 2019 *)
%o (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
%o (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 */
%o (PARI) { A060725_list(numterms) = Vec(serlaplace(exp(-x^5/5 + O(x^numterms))/(1-x))); } /* _Eric M. Schmidt_, Aug 22 2012 */
%Y Cf. A000142, A000090, A000138, A000266, A060725, A060726, A060727.
%K nonn
%O 0,3
%A Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
%E More terms from _James A. Sellers_, Apr 24 2001
%E Entry improved by comments from _Michael Somos_, Jul 28 2009
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