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A000090
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E.g.f. exp((-x^3)/3)/(1-x).
(Formerly M1295 N0496)
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9
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1, 1, 2, 4, 16, 80, 520, 3640, 29120, 259840, 2598400, 28582400, 343235200, 4462057600, 62468806400, 936987251200, 14991796019200, 254860532326400, 4587501779660800, 87162533813555200, 1743250676271104000, 36608259566534656000, 805381710463762432000
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 3-cycle.
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REFERENCES
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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.
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LINKS
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Christian G. Bower, Table of n, a(n) for n=0..100
Simon Plouffe, Exact formulas for integer sequences
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FORMULA
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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
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EXAMPLE
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a(3) = 4 because the permutations in S_3 that contain no 3-cycles are the trivial permutation and the 3 transpositions.
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MAPLE
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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
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nn=20; Range[0, nn]!CoefficientList[Series[Exp[-x^3/3]/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Oct 28 2012 *)
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PROG
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(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 */ - Entry improved by comments from Michael Somos Jul 28 2009
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CROSSREFS
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Cf. A000142, A000138, A000266, A060725.
Sequence in context: A025225 A213010 A000831 * A212432 A013115 A007171
Adjacent sequences: A000087 A000088 A000089 * A000091 A000092 A000093
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
Entry improved by comments from Michael Somos Jul 28 2009
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STATUS
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approved
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