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Search: id:A000138
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| A000138 |
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Expansion of exp (-x^4 /4) / (1-x).
(Formerly M1635 N0638)
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+0
5
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
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
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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
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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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PROGRAM
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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 */ - Entry improved by comments from Michael Somos Jul 28 2009
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CROSSREFS
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Cf. A000142, A000090.
Sequence in context: A118455 A165774 A053505 this_sequence A028857 A052687 A162059
Adjacent sequences: A000135 A000136 A000137 this_sequence 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 (njas(AT)research.att.com).
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EXTENSIONS
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Entry improved by comments from Michael Somos Jul 28 2009
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