A009837 - OEIS

%I #19 Feb 04 2022 09:53:24

%S 1,56,46336,182844416,2205641015296,63603482126974976,

%T 3748468097940723859456,404963012992964559934324736,

%U 74035562436962891333631597346816,21543523644222111882868080329093021696,9505937948357641883573662624456235995365376

%N Expansion of tanh(x)tan(x)/2 in powers of x^(4*n+2).

%H G. C. Greubel, <a href="/A009837/b009837.txt">Table of n, a(n) for n = 0..120</a>

%F a(n) = [ x^(4*n+2) ] ( tanh(x)*tan(x)/2 ).

%F a(n) = A024342(n)/2.

%F a(n) ~ (4*n+2)! * 2^(4*n+4) * tanh(Pi/2) / Pi^(4*n+3). - _Vaclav Kotesovec_, Jan 24 2015

%t nn=20; Table[(CoefficientList[Series[(Tan[x]*Tanh[x])/2,{x,0,4*nn+2}],x] * Range[0,4*nn+2]!)[[n]],{n,3,4*nn+1,4}] (* _Vaclav Kotesovec_, Jan 24 2015 *)

%o (Magma)

%o m:=50; R<x>:=PowerSeriesRing(Rationals(), m);

%o b:= Coefficients(R!(Laplace( Tan(x)*Tanh(x)/2 )));

%o [b[4*n-3]: n in [1..Floor((m-2)/4)]]; // _G. C. Greubel_, Jan 31 2022

%o (Sage) [factorial(4*n+2)*( tan(x)*tanh(x)/2 ).series(x, 4*n+3).list()[4*n+2] for n in (0..20)] # _G. C. Greubel_, Jan 31 2022

%Y Cf. A024342.

%K nonn

%O 0,2

%A _R. H. Hardin_

%E Extended and signs tested Mar 1997.