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A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

%I A005981 M4276 #48 Feb 16 2025 08:32:29
%S A005981 1,1,6,71,1456,45541,2020656,120686411,9336345856,908138776681,
%T A005981 108480272749056,15611712012050351,2664103110372192256,
%U A005981 531909061958526321421,122840808510269863827456,32491881630252866646683891,9758611490955498257378246656
%N A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.
%D A005981 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005981 P. R. Stein, personal communication.
%H A005981 Alois P. Heinz, <a href="/A005981/b005981.txt">Table of n, a(n) for n = 0..100</a>
%H A005981 Nicolas Basset, <a href="http://hal.archives-ouvertes.fr/hal-00820373/">Counting and generating permutations using timed languages</a>, 2013.
%H A005981 Nicolas Basset, <a href="https://hal.archives-ouvertes.fr/hal-01093994">Counting and generating permutations in regular classes of permutations</a>, HAL Id: hal-01093994, 2014.
%H A005981 B. Shapiro and A. Vainshtein, <a href="http://arXiv.org/abs/math.AG/0209062">Counting real rational functions with all real critical values</a>, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
%H A005981 P. R. Stein & N. J. A. Sloane, <a href="/A005981/a005981.pdf">Correspondence, 1975</a>
%H A005981 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html">Generalized Hyperbolic Functions</a>
%F A005981 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
%F A005981 Basset (2013) gives an e.g.f. involving trigonometric and hyperbolic functions. - _N. J. A. Sloane_, Dec 24 2013
%F A005981 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
%p A005981 g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%,x,35):
%p A005981 seq(n!*coeff(%,x,n),n=1..34,2); # _Peter Luschny_, Feb 07 2017
%t A005981 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 *)
%Y A005981 Bisection of A058258.
%Y A005981 Cf. A131453, A131454, A131455, A076417.
%K A005981 nonn
%O A005981 0,3
%A A005981 _N. J. A. Sloane_