"A dream" of a series :-)

[David W. Wilson]

Mr. helms effectively defines the sequence of rations c where

    e^x = PROD(n >= 1; 1 + c_n x^n)

Thus c = (1, 1/2, -1/3, 3/8, -1/5, ...), indexed starting at 1.

I observe that taking the log of this expression yields

    x = SUM(n >= 1; log(1 + c_n x^n))

If you take a Taylor series of the log expression and reorder the resulting
sum, you can obtain an expression for c_n in terms of earlier c_d where d|n.

I was just tempted to use d <| n for d is a proper divisor of n.

> -----Original Message-----
> From: Gottfried Helms [mailto:Annette.Warlich at t-online.de]
> Sent: Wednesday, June 04, 2008 9:57 AM
> To: Seqfan at ext.jussieu.fr
> Subject: "A dream" of a series :-)
> 
> A dream of a series...
> 
> Consider the exponentialseries
> 
>    E0 =  1 + x/1! + x^2/2! + x^3/3!  + ...
> 
> [&c]