# [seqfan] Identities for the Riemann zeta function gives same first

Tue Dec 16 22:54:18 CET 2008

```The polynomials of interest in the paper of M. O. Rubinstein
are the polynomials alpha_{n}(s) defined in formula (1.4).

They give rise to a sequence of integers which might
be new and interesting.

Let R(n,k) for n=0,1,.. and k=0,1,.. defined by
1
-1, 1
-2, -1, 3
-2, -1, 2, 1
-152, -78, 125, 90, 15
-216, -114, 157, 135, 35, 3
-41424, -22444, 27552, 26551, 8505, 1197, 63
-66000, -36620, 40976, 42917, 15652, 2814, 252, 9

alpha_{n}(s) = Sum(k=0..n, R(n,k) s^k) / A053657(n)

%I A000001
%S A000001 1,1,1,2,1,3,2,1,2,1,152,78,125,90,15,216,114,157,135,35,3,
%T A000001
41424,22444,27552,26551,8505,1197,63,66000,36620,40976,42917,15652,
%U A000001 2814,252,9
%V A000001
1,-1,1,-2,-1,3,-2,-1,2,1,-152,-78,125,90,15,-216,-114,157,135,35,3,
%W A000001
-41424,-22444,27552,26551,8505,1197,63,-66000,-36620,40976,42917,15652,
%X A000001 2814,252,9
%N A000001 Scaled coefficients of the M. O. Rubinstein polynomials.
%C A000001 The coefficients are in ascending order.
%D A000001 M. O. Rubinstein, Identities for the Riemann Zeta function,
arXiv:0812.2592v1 [math.NT] 14 Dec 2008.
%H A000001 <a href="http://http://arxiv.org/pdf/0812.2592v1">Identities
for the Riemann Zeta function.</a>
%F A000001 The coefficients of the polynomials alpha_{k}(s)*A053657(k) where

alpha_{0}(s) = 1 and alpha_{k+1}(s) =
(s-1)/(k+2)-sum(j=1..k,((j-(s-1)*
(k-j+1))/(k-j+2))*alpha_{j}(s))/(k+1).
%e A000001 alpha_{0}(t) = 1 / 1;
%e A000001 alpha_{1}(t) = (-1 + t) / 2;
%e A000001 alpha_{2}(t) = (-2 - t + 3t^2) / 24;
%e A000001 alpha_{3}(t) = (-2 - t + 2t^2 + t^3) / 48;
%Y A000001 A053657
%K A000001 easy,sign,tabl
%O A000001 0,4
%A A000001 Peter Luschny (peter(AT)luschny.de), Dec 16 2008

Regards
Peter Luschny

```