[seqfan] Re: [math-fun] strange approximations for F(x) = partition function. regarding A000041.

[Victor Miller]

Hi Simon, I would bet that your values are related to this:
http://www.mathstat.carleton.ca/~williams/papers/pdf/220.pdf

Victor

On Wed, Feb 23, 2011 at 12:11 PM, Simon Plouffe <simon.plouffe at gmail.com>wrote:

> Hello,
>
>  I stumbled on these 2 approximations regarding F(x),
> the partition function, where p(n) = the number
> of partitions of n, as usual. aka A000041.
>
>  F(x) = sum(p(n)*x^n, n=0..infinity):
>
>  Instead I use F*(x) = sum(p(n)*x^(n+1), n=0..infinity):
>
> Then here is the strange thing, for x = exp(-2*Pi/5) then
> the value is 1/sqrt(5), well almost ; the precision is
> 13 digits.
>
>  For x = exp(-4*Pi/5) the value is 1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5)))
> the
> precision is 28 decimal digits. I find this quite surprising.
>  I was sure it was exact, it is NOT. I verified with large values.
>
>  Also, apparently these are the only 2 examples I have found
> within F60 : the Farey fractions up to denominators = 60.
> Also when x = exp(-Pi/5) = apparently nothing algebraic of
> a low degree.
>
> caution : do not mistake these values for the standard
> F(x) which goes 0 for the exponent too, it is not the
> same.
>
>  I added these 2 values in the formulas of A000041 of course.
>
>  Does anybody have an idea why these values just pop out
> like that and apparently no other ???!
>
>  Bonne journée.
>  Simon Plouffe
>
>
>
>
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