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
Alois P. Heinz, Table of n, a(n) for n = 0..100
Nicolas Basset, Counting and generating permutations using timed languages, 2013.
Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.
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
KEYWORD
nonn
AUTHOR
STATUS
approved