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:
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Exp[-(x^5/5)]/(1-x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 24 2019 *)
PROG
(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 */
CROSSREFS
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