The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #14 Jun 11 2023 18:55:22
%S 0,0,0,2,6,18,70,310,1582,9058,57678,403878,3085478,25535378,
%T 227589206,2173314806,22137209694,239580726978,2745392996254,
%U 33207657441094,422813028038230,5652593799727858,79168165551184422,1159200449070638742,17711278225214739086
%N Expansion of e.g.f.: 2*(x-1)*tan(x/2+Pi/4)-x^2+2.
%H Y. Sano, <a href="https://doi.org/10.1016/j.disc.2006.11.019">The principal numbers of K. Saito for the types A_l, D_l and E_l</a>, Discr. Math., 307 (2007), 2636-2642.
%F E.g.f. E(x)=2*(x-1)*tan(x/2+Pi/4)-x^2+2 = 2*x - x^2 + 4*x*(x-1)/(Q(0)-x) where Q(k) = 4*k + 2 - x^2/Q(k+1); (continued fraction, 1-step).- Sergei N. Gladkovskii, Jun 22 2012
%F a(n) ~ n! * 2^(n + 2) * (Pi - 2) / Pi^(n + 1). - _Vaclav Kotesovec_, Mar 12 2019
%t With[{nn=30},CoefficientList[Series[2(x-1)Tan[x/2+Pi/4]-x^2+2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 11 2023 *)
%Y Essentially the same as 2*A034428.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Oct 30 2007
%E Definition clarified by _Harvey P. Dale_, Jun 11 2023