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A000138
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E.g.f. exp(-x^4/4)/(1-x).
(Formerly M1635 N0638)
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8
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1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000
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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 4-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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T. D. Noe, 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/4]( (-1)^i /(i! * 4^i)); a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 4^i) = e^(-1/4); a(n) ~ e^(-1/4) * n!; a(n) ~ e^(-1/4) * (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=4, n>=0. Simon Plouffe, from old notes of 1993.
E.g.f.: exp(-x^4/4)/(1-x)=1/G(0) ; G(k)= 1 - x/(1 - (x^3)/(x^3 - 4*(k+1)/G(k+1))); (continued fraction 3 kind, 3-step). - Sergei N. Gladkovskii, Feb 28 2012
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EXAMPLE
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a(4) = 18 because in S_4 the permutations with no 4-cycle are the complement of the six 4-cycles so a(4) = 4! - 6 = 18.
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MATHEMATICA
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nn=20; Range[0, nn]!CoefficientList[Series[Exp[-x^4/4]/(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^4/4) + x*O(x^n)) / (1 - x), n))} /* Michael Somos Jul 28 2009 */
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CROSSREFS
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Cf. A000142, A000090, A000266, A060725, A060726, A060727
Sequence in context: A118455 A165774 A053505 * A028857 A052687 A162059
Adjacent sequences: A000135 A000136 A000137 * A000139 A000140 A000141
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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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Entry improved by comments from Michael Somos, Jul 28 2009.
Name corrected by Joerg Arndt, May 27 2011.
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STATUS
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approved
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