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

[Brendan McKay]

Hi Christian, This example shows the importance of monitoring rounding
error.  I'm not confident that any of the algebra systems people are using
can be trusted to monitor it on their own.

Your value:
-4.44582 97365 50756 88210 15903 52124 64363 74114 55901 97524 36514 54 E-39
Correct value:
-4.44582 97365 50756 88210 15903 52124 64363 74014 36685 74871 82886 70 E-39

I.e., only 38 digits are correct.

Brendan.

On 19/5/2023 3:42 pm, Christian Sievers wrote:
> 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
>
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