A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x). (Formerly M1635 N0638)

1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000

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.

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Simon Plouffe, Exact formulas for integer sequences

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

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

Sequence in context: A118455 A165774 A053505 * A028857 A052687 A302546

Adjacent sequences: A000135 A000136 A000137 * A000139 A000140 A000141

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Entry improved by comments from Michael Somos, Jul 28 2009

Name corrected by Joerg Arndt, May 27 2011

Status

approved