A109577 - OEIS

%I #8 Sep 26 2013 03:44:45

%S 0,1,0,3,-12,65,-450,3577,-32424,331137,-3757350,46892681,-638447436,

%T 9416929249,-149581289130,2545707159465,-46213451575248,

%U 891368532889601,-18204123410896590,392431030262264329,-8904987308885931060,212174197452256551393,-5296088301994320448530

%N E.g.f.: 2x/[1-exp(-2x)+2exp(-x)].

%F E.g.f.: (2*x)/(1-exp(-2*x)+2*exp(-x))=2*x/(1 +2*G(0)) ; G(k)= 1-(2^k)/(2-4*x/(2*x+(2^k)*(k+1)/G(k+1))); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 08 2012

%F a(n) ~ n! * (-1)^(n+1) * (1-1/sqrt(2))/log(1+sqrt(2))^n. - _Vaclav Kotesovec_, Sep 26 2013

%p G:=2*x/(1-exp(-2*x)+2*exp(-x)): Gser:=series(G,x=0,26): 0,seq(n!*coeff(Gser,x^n),n=1..23); # yields the signed sequence

%t g[x_] = x/(-1 + Exp[ -2*x] - 2*Exp[ -x]) h[x_, n_] = Dt[g[x], {x, n}] a[x_] = Table[h[x, n], {n, 0, 50}]; b = 2*a[0]

%K sign

%O 0,4

%A _Roger L. Bagula_, Jun 28 2005