[seqfan] Re: A063747 sign of power series expansion of the Gamma function

[Christian Sievers]

On Fri, May 19, 2023 at 02:31:10PM +1200, Sean A. Irvine wrote:

> Hi,
> 
> I believe the terms in A063747 are incorrect for n>=46.
> 
> https://oeis.org/A063747
> 
> My attempts to reproduce the calculation in Maple and using constructible
> reals in Java both yield a coefficient, like -0.44458...e-38 * x^46, from
> which a(46) should be -1 rather than 1.
> 
> Is someone able to make an independent check?
> 
> I was unable to run the existing Pari program beyond n=15, but I assume
> there is some way to tell Pari to try harder?

The O-Term needs to be in the argument to gamma.
With that we get the result of one.

But with a higher precision, PARI confirms your result:

? \p60
   realprecision = 77 significant digits (60 digits displayed)
? polcoef(1/gamma(1-x+O(x^47)),46)
%1 = -4.445829736550756882101590352124643637411455901975243651454 E-39

(polcoeff is deprecated in favour of polcoef.)


All the best,
Christian