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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
| 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
| Christian G. Bower, Table of n, a(n) for n=0..100
Plouffe Simon, Exact formulas for integer sequences
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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
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EXAMPLE
| 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
| 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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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 */ - Entry improved by comments from Michael Somos Jul 28 2009
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CROSSREFS
| Cf. A000142, A000138, A000266, A060725.
Sequence in context: A115125 A025225 A000831 * A013115 A007171 A058136
Adjacent sequences: A000087 A000088 A000089 * A000091 A000092 A000093
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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