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A003483
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Number of square permutations of n elements.
(Formerly M2931)
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8
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1, 1, 1, 3, 12, 60, 270, 1890, 14280, 128520, 1096200, 12058200, 139043520, 1807565760, 22642139520, 339632092800, 5237183952000, 89032127184000, 1475427973219200, 28033131491164800, 543494606861606400
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of permutations p in S_n such that there exists q in S_n with q^2=p.
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REFERENCES
| Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, 485-515.
J. Blum, Enumeration of the square permutations in S_n, J. Combin. Theory, A 17 (1974), 156-161.
Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370
N. Pouyanne, On the number of permutations admitting an m-th root, Electron. J. Combin., 9 (2002), #R3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.11.
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LINKS
| P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148, Eq. 4.8.1.
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FORMULA
| E.g.f.: sqrt((1+x)/(1-x))*Product_{k >= 1} cosh x^(2k)/(2k) [Blum, corrected].
a(2n+1)=(2n+1)a(2n).
Asymptotics: a(n) ~ n! 2/sqrt(n pi) e^G, where e^G = prod_{k>=1} cosh(1/(2k)) ~ 1.22178
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EXAMPLE
| a(3)=3: permutations with square roots are identity and two 3-cycles.
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MATHEMATICA
| max = 20; f[x_] := Sqrt[(1 + x)/(1 - x)]* Product[ Cosh[x^(2*k)/(2*k)], {k, 1, max}]; se = Series[ f[x], {x, 0, max}]; CoefficientList[ se, x]*Range[0, max]! (* From Jean-François Alcover, Oct 05 2011, after g.f. *)
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CROSSREFS
| Cf. A103619 (cube root), A103620 (fourth root).
Sequence in context: A127918 A069944 A073996 * A128602 A092803 A181282
Adjacent sequences: A003480 A003481 A003482 * A003484 A003485 A003486
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 28 2001
Additional comments from Michael Somos, Jun 27, 2002
It would be nice to have cross-references to number of permutations admitting a cube root, etc.! - N. J. A. Sloane (njas(AT)research.att.com), Jan 11 2005
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