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

[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
 sequences, which might be understood as a0(n) "of order 0",
 a1(n) of "order 1" -  and I shall also later (by generalization
 of the computations) be confronted with more sequences a2(n),
 a3(n), a4(n),... similar/related to the first two ones, but
 just of "higher orders" - but don't have that coefficients yet.

 The sequence a0(0) is obvious - this is the sequence of
 zeta-values at nonpositive arguments.
 I suspect, the sequence a1(n) is somehow a convolution, or
 transformation of that but do not have any really convincing idea;
 however in the third column I compute 2*a1(k) + a0(k-1) and find
 at least, that each second one is exactly a bernoulli-number.

 Here is the table of sequences:

            a0(k)          a1(k)            2 a1(k) + a0(k-1)
--------------------------------------------------------------
            0                .                       .
            0                0                       .
         -1/2                0                       0
        -1/12              1/4                       0
            0              1/8                        1/6
        1/120             1/48                    1/24
            0            -1/48                       -1/30
       -1/252            -1/96                   -1/48
            0             1/72                        1/42
        1/240         101/8640                101/4320
            0           -3/160                        -1/30
       -1/132          -13/576                 -13/288
            0             1/24                         5/66
    691/32760         402/6079                804/6079
            0        -691/5040                        -691/2730
        -1/12       -2357/8640              -2357/4320
            0              5/8                         7/6

The sequences are taken by approximation of lists of coefficients,
which I find by a linear regression based on data which must be
computed by a very time-consuming procedure with high precision.

The sequence of regression-coefficients a1(n) is thus basically a
sequence of real numbers, but approximations to rational numbers with
at most 4-digit-denominators give then the above guesses.
Because of the time and precision required for the computation of the
underlying data I unfortunately cannot easily produce more
(somehow reliable) such sequence-entries.

For that found entries in s1() I'm pretty sure that that values are
the most plausible ones, but single ones with the higher denominators
might be incorrect guesses.


So does someone have an idea, how possibly a1(n) could be composed
by the bernoulli-numbers/zetas-at-negative-arguments?

Gottfried

----------------------------------------------------------------------

[*1] testing number-theoretical properties of my summation-method
with the matrix of factorially scaled Eulerian numbers for summation
  * of 1/k^2 (arriving at a0(n)) and
  * of 1/k^3 (arriving at a1(n)).


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