A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1. (Formerly M4276)

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

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

Bisection of A058258.

Cf. A131453, A131454, A131455, A076417.

Sequence in context: A127135 A187651 A357141 * A024272 A167813 A242232

Adjacent sequences: A005978 A005979 A005980 * A005982 A005983 A005984

Keyword

nonn

Author

N. J. A. Sloane

Status

approved