[seqfan] Re: Unknown sequence related to Bernoulli-numbers/zeta() at negative arguments / solution

[Gottfried Helms]

Am 21.06.2014 12:47 schrieb Gottfried Helms:
> Am 21.06.2014 08:16 schrieb Paul D Hanna:
>> Hi Gottfried,
>>     You were correct in that it a0() and a1() are related by convolution.
(...)
> Hi Seqfans, hi Paul -
> 
>  your idea was perfect. It is immediately generalizable to all sequences a_k(n).
>  I'll show how this can be done:
> 
That idea of Paul brought me to the complete solution of
the question behind. I was studying properties of the
transformation by the matrix of Eulerian numbers (which
is also rescaled by factorials), so is one of the sequence-
transformations which is also possibly generally interesting
for the OEIS-database, btw.

The coefficients for which I sought the generating scheme
occur, when the said Eulerian transformation was applied
to sequences, which would give nonsummable series, so
for some type of series (nonalternating divergent) which
cannot be summed by the according "Eulerian summation" -
the sought coefficients let us determine a matrix of
systematic error-terms when divergent Dirichlet/Zeta-series
are fed in that Eulerian transformation.

I don't know whether this is really interesting for the
OEIS or SeqFans, but if it is, here is the link to
my finished study of the Eulerian transformation in cases of
non-summable divergence where that coefficients appear
in chap 3 and paragraph 3.3.2 (pg 14):

  http://go.helms-net.de/math/binomial_new/EulerianSumsV2.pdf

Gottfried