A131281 Expansion of e.g.f.: 2*(x-1)*tan(x/2+Pi/4)-x^2+2.

0, 0, 0, 2, 6, 18, 70, 310, 1582, 9058, 57678, 403878, 3085478, 25535378, 227589206, 2173314806, 22137209694, 239580726978, 2745392996254, 33207657441094, 422813028038230, 5652593799727858, 79168165551184422, 1159200449070638742, 17711278225214739086

Offset: 0

Author

N. J. A. Sloane, Oct 30 2007

Keywords

• nonn

Links

• Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.

Crossrefs

Essentially the same as 2*A034428.

Programs

Mathematica

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 *)

Formula

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

a(n) ~ n! * 2^(n + 2) * (Pi - 2) / Pi^(n + 1). - Vaclav Kotesovec, Mar 12 2019

Extensions

Definition clarified by Harvey P. Dale, Jun 11 2023