A060727 For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.

1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 75613824000, 1134207360000, 18147317760000, 308504401920000, 5553079234560000, 105508505456640000, 2110170109132800000, 44288746761093120000, 974352428744048640000

Offset

0,3

Comments

This is the expansion of exp ((-x^7)/7)/(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

Formula

The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/7 ]( (-1)^i /(i! * 7^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 7^i) = e^(-1/7) or a(n) ~ e^(-1/7) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/7) * (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=7, n>=0. - Simon Plouffe, Feb 18 2011

Example

a(7) = 4320 because in S_7 the permutations with no 7-cycle are the complement of the 720 7-cycles so a(7) = 7! - 720 = 4320.

Maple

for n from 0 to 30 do printf(`%d, `, n! * sum(( (-1)^i /(i! * 7^i)), i=0..floor(n/7))) od:

Prog

(PARI) { for (n=0, 100, write("b060727.txt", n, " ", n! * sum(i=0, n\7, (-1)^i / (i! * 7^i))); ) } \\ Harry J. Smith, Jul 10 2009

Crossrefs

Cf. A000142, A000090, A000266, A000138, A060725, A060726, A060727.

Sequence in context: A070947 A267386 A215718 * A152350 A152368 A152364

Adjacent sequences: A060724 A060725 A060726 * A060728 A060729 A060730

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