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A005981
Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.
(Formerly M4276)
9
1, 1, 6, 71, 1456, 45541, 2020656, 120686411, 9336345856, 908138776681, 108480272749056, 15611712012050351, 2664103110372192256, 531909061958526321421, 122840808510269863827456, 32491881630252866646683891, 9758611490955498257378246656
OFFSET
0,3
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.
LINKS
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
P. R. Stein & N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions
FORMULA
E.g.f.: x + Sum_{n>=1} a(n)*(x^(2n+1))/(2n+1)! = (f(0,x)*f(1,x) -f(2,x)*f(3,x)+ f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = Sum_{k>=0} (x^(4k+j))/(4k+j)!, j = 0,1,2,3, is the j-th generalized hyperbolic function. - Peter Bala, Jul 13 2007
Basset (2013) gives an e.g.f. involving trigonometric and hyperbolic functions. - N. J. A. Sloane, Dec 24 2013
a(n) ~ 4 * (2*n+1)! / (tan(r/2)^2 * r^(2*n+2)), where r = A076417 = 1.8751040687119611664453082410782141625701117335310699882454137131... is the root of the equation cos(r)*cosh(r) = -1. - Vaclav Kotesovec, Feb 01 2015
MAPLE
g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%, x, 35):
seq(n!*coeff(%, x, n), n=1..34, 2); # Peter Luschny, Feb 07 2017
MATHEMATICA
egf = ((Cosh[x]-1)*Sin[x]+(Cos[x]+1)*Sinh[x])/(Cos[x]*Cosh[x]+1); a[n_] := SeriesCoefficient[egf, {x, 0, 2*n+1}]*(2*n+1)!; Array[a, 17, 0] (* Jean-François Alcover, Mar 13 2014 *)
CROSSREFS
Bisection of A058258.
Sequence in context: A127135 A187651 A357141 * A024272 A167813 A242232
KEYWORD
nonn
STATUS
approved