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Search: id:A000266
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| A000266 |
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Expansion of exp (-x^2 /2) / (1-x).
(Formerly M2991 N1211)
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+0
9
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| 1, 1, 1, 3, 15, 75, 435, 3045, 24465, 220185, 2200905, 24209955, 290529855, 3776888115, 52876298475, 793144477125, 12690313661025, 215735332237425, 3883235945814225, 73781482970470275, 1475629660064134575
(list; graph; listen)
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OFFSET
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0,4
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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 transposition.
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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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 104
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FORMULA
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a(n) = n! * sum i=0 ... [n/2]( (-1)^i /(i! * 2^i)); a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 2^i) = e^(-1/2); a(n) ~ e^(-1/2) * n!; a(n) ~ e^(-1/2) * (n/e)^n * sqrt(2 * Pi * n). - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
A027616(n) + a(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
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EXAMPLE
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a(3) = 3 because the permutations in S_3 that contain no transpositions are the trivial permutation and the two 3-cycles.
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^2 / 2) + 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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Sequence in context: A002902 A005053 A136778 this_sequence A059838 A079164 A047015
Adjacent sequences: A000263 A000264 A000265 this_sequence A000267 A000268 A000269
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KEYWORD
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nonn
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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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More terms from Christian G. Bower (bowerc(AT)usa.net).
Entry improved by comments from Michael Somos Jul 28 2009
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