eulerian numbers for negative argmuents

[Gottfried Helms]

Am 15.03.2006 17:47 schrieb Mitch Harris:
> Thanks for the lead. If I can't get a (bivariate) recurrence out of it, 
> at least there's a summation formula (which is opposite of what one 
> usually wants).
> 
Mitch,

 the recurrence is similar to the pascal-triangle .

  1  2  3 4   5  col
 ---------------+-------------
  1             | row 1
  1 1           |     2
  1 4  1        |     3
  1 11 11 1     |     4
  1 26 66 26  1 |     5

  Example row 4 is constructed from row 3

   1 =  4*0   +  1*1
   11=  3*1   +  2*4
   11=  2*4   +  3*1
   1 =  1*1   +  4*0


  Example row 5 is constructed from row 4

   1 =  5*0   +  1*1
   26=  4*1   +  2*11
   66=  3*11  +  3*11
   26=  2*11  +  4*1
   1 =  1*1   +  5*0

 and so on.

Regards -

Gottfried Helms