A000090 Expansion of e.g.f. exp((-x^3)/3)/(1-x). (Formerly M1295 N0496)

1, 1, 2, 4, 16, 80, 520, 3640, 29120, 259840, 2598400, 28582400, 343235200, 4462057600, 62468806400, 936987251200, 14991796019200, 254860532326400, 4587501779660800, 87162533813555200, 1743250676271104000, 36608259566534656000, 805381710463762432000

Offset

0,3

Comments

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 3-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

Christian G. Bower, Table of n, a(n) for n = 0..100

Simon Plouffe, Exact formulas for integer sequences

L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976

Formula

a(n) = n! * Sum_{i=0..floor(n/3)} (-1)^i / (i! * 3^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 3^i) = e^(-1/3); a(n) ~ e^(-1/3) * n!; a(n) ~ e^(-1/3) * (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=3, n>=0. - Simon Plouffe from old notes, 1993

E.g.f.: E(x) = exp(-x^3/3)/(1-x)=G(0)/((1-x)^2); G(k) = 1 - x/(1 - x^2/(x^2 + 3*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 11 2012

Example

a(3) = 4 because the permutations in S_3 that contain no 3-cycles are the trivial permutation and the 3 transpositions.

Maple

seq(coeff(convert(series(exp((-x^3)/3)/(1-x), x, 50), polynom), x, i)*i!, i=0..30); # series expansion A000090:=n->n!*add((-1)^i/(i!*3^i), i=0..floor(n/3)); seq(A000090(n), n=0..30); # formula (Pab Ter)

Mathematica

nn=20; Range[0, nn]!CoefficientList[Series[Exp[-x^3/3]/(1-x), {x, 0, nn}], x]  (* Geoffrey Critzer, Oct 28 2012 *)

Prog

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^3 / 3) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */

Crossrefs

Cf. A000142, A000138, A000266, A060725.

Sequence in context: A326859 A213010 A000831 * A295922 A300100 A212432

Adjacent sequences: A000087 A000088 A000089 * A000091 A000092 A000093

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Entry improved by comments from Michael Somos, Jul 28 2009

Status

approved