|
%I #20 Sep 08 2022 08:45:16
%S 1,1,5,5,-5815,-956375,-172917875,-38579649875,-10713341611375,
%T -3663118565923375,-1519935859717136875,-754429769289426936875,
%U -442113820341129750734375,-302333022017412857174234375,-238762676857713027642764171875,-215766282905942334008224968671875
%N Expansion of e.g.f. sin(arctanh(x)), odd powers only.
%H Vincenzo Librandi, <a href="/A102058/b102058.txt">Table of n, a(n) for n = 1..100</a>
%F a(n) = (2*n-1)!*Sum(i=0..n-1, (-1)^(i)*Sum(k=0..2*n-1-2*i-1, (stirling1(k+2*i+1,2*i+1)*2^(k)* binomial(2*n-2,k+2*i))/(k+2*i+1)!)). - _Vladimir Kruchinin_, Dec 12 2011
%e sin(arctanh(x)) = x + x^3/3! + 5x^5/5! + 5x^7/7! - 5815x^9/9! - ...
%t nmax=20; Table[(CoefficientList[Series[Sin[ArcTanh[x]],{x,0,2*nmax}],x] * Range[0,2*nmax-1]!)[[n]],{n,2,2*nmax,2}] (* _Vaclav Kotesovec_, Nov 06 2014 *)
%o (Maxima)
%o a(n):=(2*n-1)!*sum((-1)^(i)*sum((stirling1(k+2*i+1,2*i+1)*2^(k)* binomial(2*n-2,k+2*i))/(k+2*i+1)!,k,0,2*n-1-2*i-1),i,0,n-1); /* _Vladimir Kruchinin_, Dec 12 2011 */
%o (Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1+Sin(Argtanh(x)))); [Factorial(n-1)*b[n]: n in [2..m by 2]]; // _Vincenzo Librandi_, Aug 16 2018
%Y Bisection of A002019.
%K sign
%O 1,3
%A _Ralf Stephan_, Dec 28 2004
|
