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A007999
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a(n) = number of permutations w of 1,2,...,n such that both w and w^{-1} are alternating.
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1
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1, 1, 2, 3, 8, 19, 64, 213, 880, 3717, 18288, 92935, 531440, 3147495, 20525168, 138638825, 1015694832, 7700244745, 62623847536, 526317901451, 4705365925872, 43407723925499, 423149546210416, 4250149857500861, 44868038386273776
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..24.
H. O. Foulkes, Tangent and secant numbers and representations of symmetric groups, Discrete Math. 15 (1976), no. 4, 311-324.
R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006. [Joel B. Lewis, May 21 2009]
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FORMULA
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sum_{n=0..infinity} a(n)x^n = sum_{k=0..infinity} E_{2k+1}^2 u^{2k+1}/(2k+1)! + (1-x^2)^{-1/2} sum_{k=0..infinity} E_{2k}^2 u^{2k}/(2k)!, where E_j is an Euler number and u = (1/2)log((1+x)/(1-x)). - Richard Stanley, Jan 21 2006
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MATHEMATICA
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m = 27;
e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)(2^(n+1)-1)*BernoulliB[ n+1])/(n+1)]];
u[x_] := Log[(1+x)/(1-x)]/2;
Sum[e[2k+1]^2 u[x]^(2k+1)/(2k+1)!, {k, 0, m}] + (1-x^2)^(-1/2) Sum[e[2k]^2* u[x]^(2k)/(2k)!, {k, 0, m}] + O[x]^m // Drop[CoefficientList[#, x], 2]& (* Jean-François Alcover, Feb 24 2019 *)
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CROSSREFS
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Sequence in context: A243791 A243335 A303835 * A006609 A005663 A112834
Adjacent sequences: A007996 A007997 A007998 * A008000 A008001 A008002
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KEYWORD
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nonn
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AUTHOR
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poirier(AT)lacim.uqam.ca, Simon Plouffe
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EXTENSIONS
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More terms from Vladeta Jovovic, May 15 2007
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STATUS
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approved
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