A191720 Expansion of e.g.f. arctan(exp(x)-1).

1, 1, -1, -11, -25, 181, 2039, 3109, -131545, -1417259, 2734679, 239834629, 2217084935, -23659572299, -859070164681, -5378041927451, 198832696957415, 5164126517703061, 4868884057959959, -2309488856960067131

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for n = 1..200

Eric Weisstein's MathWorld, Polylogarithm.

Formula

a(n) = sum(m=1..(n+1)/2, ((-1)^(m-1)*(2*m-2)!*stirling2(n,2*m-1))).

a(n) ~ (n-1)! * 2^(2*n) * sin(n*arctan(Pi/log(4))) / (Pi^2+4*log(2)^2)^(n/2). - Vaclav Kotesovec, Jan 02 2014

a(n) = (-1)^(n+1)*Im(Li_{1-n}(1+i)), where Li_n(x) is the polylogarithm, i=sqrt(-1). - Vladimir Reshetnikov, Oct 31 2015

Maple

a:=series(arctan(exp(x)-1), x=0, 21): seq(n!*coeff(a, x, n), n=0..20); # Paolo P. Lava, Mar 27 2019

Mathematica

nn=20; Rest[CoefficientList[Series[ArcTan[Exp[x]-1], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 22 2011 *)
Table[(-1)^(n+1) Im[PolyLog[1-n, 1+I]], {n, 1, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

Prog

(Maxima)
a(n):=sum(((-1)^(m-1)*(2*m-2)!*stirling2(n, 2*m-1)), m, 1, (n+1)/2);

Crossrefs

Sequence in context: A043160 A043940 A220434 * A066956 A250611 A137015

Adjacent sequences: A191717 A191718 A191719 * A191721 A191722 A191723

Keyword

sign

Author

Vladimir Kruchinin, Jun 13 2011

Status

approved