A000090 Expansion of e.g.f. exp((-x^3)/3)/(1-x).
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
Examples
a(3) = 4 because the permutations in S_3 that contain no 3-cycles are the trivial permutation and the 3 transpositions.
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
Programs
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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)
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Mathematica
nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^3/3]/(1-x),{x,0,nn}],x] (* Geoffrey Critzer, Oct 28 2012 *) -
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 */
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
Extensions
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
Entry improved by comments from Michael Somos, Jul 28 2009
Comments