A191719 Expansion of e.g.f. arctan(x*exp(x)).

0, 1, 2, 1, -20, -151, -354, 6217, 100472, 537777, -7631270, -223395919, -2120164188, 22050300505, 1154262915638, 17130776734905, -105423782758544, -11372993234072863, -245877012220234446, 345837436238423521, 188329590656514108380

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..100

Formula

a(n) = n!*Sum_{m=1..(n+1)/2} ((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!.

a(n) ~ (n-1)! * sin(n*arctan(1/tan(r))) * (cos(r)/r)^n, where r = Im(LambertW(I)) = A305200 = 0.576412723031435283148289239887... is the root of the equation exp(r*tan(r))=cos(r)/r. - Vaclav Kotesovec, Jan 02 2014

Mathematica

Rest[CoefficientList[Series[ArcTan[x*Exp[x]], {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 02 2014 *)

Prog

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

Crossrefs

Cf. A009635, A216401, A297009.

Sequence in context: A013158 A012932 A013163 * A009768 A051492 A380220

Adjacent sequences: A191716 A191717 A191718 * A191720 A191721 A191722

Keyword

sign

Author

Vladimir Kruchinin, Jun 13 2011

Extensions

a(0)=0 prepended by Seiichi Manyama, Oct 01 2021

Status

approved