A000834 Expansion of exp(x)*(1 + tan(x))/(1 - tan(x)).

1, 3, 9, 35, 177, 1123, 8569, 76355, 777697, 8911683, 113466729, 1589173475, 24280777617, 401898209443, 7163977596889, 136821894075395, 2787312733887937, 60331585563062403, 1382698089425999049

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• R. J. Mathar, Table of n, a(n) for n = 0..161
• C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).

Programs

Maple

A000834 := exp(x)*(sin(x)+cos(x))/(cos(x)-sin(x)) : for n from 0 to 200 do printf("%d %d ",n,n!*coeftayl(A000834,x=0,n)) ; end: # R. J. Mathar, Nov 19 2006
A000834 := proc(n) local i; add((-1)^(i*(i-1)/2)*4^i*binomial(n,i)*(euler(i,1/2)+euler(i,1)),i=0...n)-1 end; # Peter Luschny, Nov 25 2010

Mathematica

With[{nn=20},CoefficientList[Series[Exp[x] (1+Tan[x])/(1-Tan[x]), {x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Sep 08 2011 *)

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*A000831(n-i). - R. J. Mathar, Nov 19 2006

a(n) := -1 + Sum_{i=0...n} ((-1)^(i(i-1)/2) 4^i C(n,i)(E_{i}(1/2) + E_{i}(1))), where E_{n}(x) are Euler polynomials. - Peter Luschny, Nov 25 2010

G.f.: G(0)*2*x/(1 - x)/(1 - 3*x) + 1/(1 - x), where G(k) = 1 - 2*x^2*(k+1)*(k+2)/(2*x^2*(k+1)*(k+2) - (2*x*k + 3*x - 1)*(2*x*k + 5*x - 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 24 2014

a(n) ~ n! * exp(Pi/4) * 2^(2*n+2) / Pi^(n+1). - Vaclav Kotesovec, Jul 02 2015