login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

A000138
Expansion of e.g.f. exp(-x^4/4)/(1-x).
(Formerly M1635 N0638)
9
1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000, 39789649046959812000, 875372279033115864000
OFFSET
0,3
COMMENTS
a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 4-cycle.
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
FORMULA
a(n) = n! * Sum_{i=0..floor(n/4)} (-1)^i / (i! * 4^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 4^i) = e^(-1/4); a(n) ~ e^(-1/4) * n!; a(n) ~ e^(-1/4) * (n/e)^n * sqrt(2*Pi*n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), here k=4, n>=0. Simon Plouffe, from old notes, 1993
E.g.f.: exp(-x^4/4)/(1-x) = 1/G(0); G(k) = 1 - x/(1 - (x^3)/(x^3 - 4*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 28 2012
EXAMPLE
a(4) = 18 because in S_4 the permutations with no 4-cycle are the complement of the six 4-cycles so a(4) = 4! - 6 = 18.
MATHEMATICA
nn=20; Range[0, nn]!CoefficientList[Series[Exp[-x^4/4]/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Oct 28 2012 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^4/4) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
Entry improved by comments from Michael Somos, Jul 28 2009
Name corrected by Joerg Arndt, May 27 2011
STATUS
approved