A105752 Expansion of e.g.f. cos(i*log(1 + x)), i = sqrt(-1).

1, 0, 1, -3, 12, -60, 360, -2520, 20160, -181440, 1814400, -19958400, 239500800, -3113510400, 43589145600, -653837184000, 10461394944000, -177843714048000, 3201186852864000, -60822550204416000, 1216451004088320000, -25545471085854720000, 562000363888803840000

Offset: 0

Author

Paul Barry, Apr 18 2005

Keywords

• easy
• sign

Comments

If the signs are ignored, this is essentially the same as A001710, whose e.g.f. is cos(i*log(1 - x)) = cosh(log(1 - x)).

The sequence 0,1,1,3,12,60,... has e.g.f. -Im(sin(i*log(1 - x))) = -sinh(log(1 - x)); the sequence 0,1,-1,3,-12,60,... has e.g.f. Im(sin(i*log(1 + x))) = sinh(log(1 + x)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001710.

Programs

Mathematica

CoefficientList[Series[1/2*(1+x+1/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)

PARI

x='x+O('x^66); Vec(serlaplace(1/2*(1+x+1/(1+x)))) \\ Joerg Arndt, May 15 2013

Formula

E.g.f.: cos(i*log(1 + x)), i = sqrt(-1).

E.g.f.: 1/2*(1 + x + 1/(1 + x)). - Sergei N. Gladkovskii, May 15 2013

Let Q(k,x) = 1 + (k+2)*x/(1 - x/(x + 1/Q(k+1,x))), then g.f.: 1 + (Q(0,sqrt(-x)) - 1)*x^2/(2*(sqrt(-x) - x)). - Sergei N. Gladkovskii, May 15 2013

G.f.: 1 + x^2/2*G(0), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

For n > 1, a(n) = (-1)^n * n! / 2. - Vaclav Kotesovec, Feb 25 2014

Conjecture: a(n) = Sum_{k=0..n} Stirling1(n, 2*k). - Benedict W. J. Irwin, Oct 19 2016

E.g.f.: cosh(log(1 + x)). - Jianing Song, Apr 06 2019