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 A000090 E.g.f. exp((-x^3)/3)/(1-x). (Formerly M1295 N0496) 12
 1, 1, 2, 4, 16, 80, 520, 3640, 29120, 259840, 2598400, 28582400, 343235200, 4462057600, 62468806400, 936987251200, 14991796019200, 254860532326400, 4587501779660800, 87162533813555200, 1743250676271104000, 36608259566534656000, 805381710463762432000 (list; graph; refs; listen; history; text; internal format)
 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 ... [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, 3-step ). - 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 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

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)