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%I #42 Sep 29 2023 12:01:24
%S 1,3,9,35,177,1123,8569,76355,777697,8911683,113466729,1589173475,
%T 24280777617,401898209443,7163977596889,136821894075395,
%U 2787312733887937,60331585563062403,1382698089425999049
%N Expansion of exp(x)*(1 + tan(x))/(1 - tan(x)).
%H R. J. Mathar, <a href="/A000834/b000834.txt">Table of n, a(n) for n = 0..161</a>
%H C. K. Cook, M. R. Bacon, and R. A. Hillman, <a href="https://www.fq.math.ca/Papers1/55-3/CookBaconHillman01222017.pdf">Higher-order Boustrophedon transforms for certain well-known sequences</a>, Fib. Q., 55(3) (2017), 201-208.
%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%F a(n) = Sum_{i=0..n} binomial(n,i)*A000831(n-i). - _R. J. Mathar_, Nov 19 2006
%F 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
%F 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
%F a(n) ~ n! * exp(Pi/4) * 2^(2*n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Jul 02 2015
%p 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
%p 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
%t 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 *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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