decode sequence

[Gottfried Helms]

Dear seqfans -

I have a difficult sequence, not in OEIS:

let  n begin at 1, then I have the sequence:

a(n)= 1/2  -1/12  1/48  -1/180  11/8640  -1/6720  -11/241920  29/1451520
   493/43545600  -2711/239500800  -6203/3592512000  2636317/373621248000
 -10597579/10461394944000  -439018457/78460462080000 ...

if I rescale

     b(n) = a(n)*n*n! *(n+1)!

I seem to get integers:

b(n)=  1     -1     3     -16     110      -540       -9240      292320
    14908320      -1639612800      -33013854720        21046667685120
  -549927873855360          -637881314775344640

another rescaling, perhaps a bit more smooth, is

     c(n) = a(n)*((n+1)!)^2

c(n)= 2     -3    12     -80     660     -3780      -73920     2630880
    149083200    -18035740800       -396166256640  273606679906560
    -7698990233975040      -9568219721630169600   ...

I can create this series by matrix-logarithm and exponentiation
to some finite extent, but to discuss things analytically it would
be good to have a more direct description.

Someone an idea?


Context:
It occurs in the core of a tetration-formula, where

   f_s:  x:= s^x - 1

and f_s is iterated. The special interest of these coefficients is,
that they allow to define a powerseries in y,x,log(s) for fractional
iterates of f_s, where y denotes the fractional value for the iteration.


Gottfried