OFFSET
0,3
COMMENTS
This is the expansion of exp ((-x^6)/6) /(1-x).
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..floor(n/6)} ((-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ Sum_{i>= 0} (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (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=6, n >= 0. - Simon Plouffe, Feb 18 2011
EXAMPLE
a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.
MAPLE
for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:
PROG
(PARI) a(n)={n! * sum(i=0, n\6, (-1)^i / (i! * 6^i))} \\ Harry J. Smith, Jul 10 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
EXTENSIONS
More terms from James A. Sellers, Apr 24 2001
STATUS
approved