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

[Christian Sievers]

Hi,

I gave the python + flint approach some more time and was able to
compute 10000 terms in under 10 minutes.

(As arguments to "good" I used prec=100000 so that it starts the
computation with a high precision (this is generally not a good idea
since it mainly specifies the precision demanded from the result,
but that doesn't matter in this special case), no maxprec (it gets a
high enough value as a result of the big prec), and verbose=1 in order
to see that the result comes from the first computation.)

Seeing all this +1/-1 numbers, I thought about run length encoding them.
Since there are only two values, it is enough to know (the first value
and) the run lengths alone. Among the values I looked at, the lengths
were at most 9. So it seemed natural to just append all the digits.
I find the result quite impressive:
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Note that the last value represents an incomplete run.
All other values differ by at most 1 from the previous,
and after a value n is reached, we never get back to n-2.

Maybe this or the general appearance reflects something that is well
known about these coefficients, but maybe not.

Doing the same with the mostly wrong first 1000 results that PARI gives
with default precision gives a very different picture, it ends with
132123221321223122312141211122111121211121214121112211412111212132214.


Bye,
Christian