# [seqfan] Re: Unknown sequence related to Bernoulli-numbers/zeta() at n egative arguments

Gottfried Helms helms at uni-kassel.de
Sat Jun 21 09:54:18 CEST 2014

```Hi Paul -

wow. That's a surprise - a0() and a1() so tightly related! Thank you very much!

I should then look what this means for the coefficients without the
additional factorials, maybe there is something in that manner.
here is a link, where I discuss the guesses for a0() and find some
interesting stuff with the psi()/Digamma()-function and a series
of products of zetas like zeta(0)*zeta(2) + zeta(-1)*zeta(3)+...
see here for more:

http://math.stackexchange.com/questions/839679

Thank you again -

cordially -
Gottfried!

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.
> Define
>  A1 = 1/4*x^2/2! + 1/8*x^3/3! + 1/48*x^4/4! - 1/48*x^5/5! - 1/96*x^6/6! + 1/72*x^7/7! + 101/8640*x^8/8! - 3/160*x^9/9! - 13/576*x^10/10! + 1/24*x^11/11! + 7999/120960*x^12/12! - 691/5040*x^13/13! - 2357/8640*x^14/14! + 5/8*x^15/15! + 52037/34560*x^16/16! +...thensqrt(2*A1) = log( (exp(x)-1)/x )   = 1/2*x + 1/12*x^2/2! - 1/120*x^4/4! + 1/252*x^6/6! - 1/240*x^8/8! + 1/132*x^10/10! +...
>  .
> Note that your a1 term  402/6079  should have been  7999/120960 ...
>
> Regards,
>    Paul
>
> ---------- Original Message ----------
> From: Gottfried Helms <helms at uni-kassel.de>
> To: "M <SeqFanList>" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Unknown sequence related to Bernoulli-numbers/zeta() at negative arguments
> Date: Sat, 21 Jun 2014 04:37:56 +0200
>
> Hi -
>  analyzing some matrix-summation-scheme [*1] I come across two

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