Summation proof

[John Keith]

To clarify my (rather rusty) recollection, the LHS reads "the derivative of COMB(n, k) where x = 0"? Also, can you explain the connection to Bernoulli numbers?

Empirically, here is a table of your function for n from 0 to 7 and k from 0 to n.

Code:


 { Inf }
 { '3/2'   1 }
 { '5/6'   1 '3/2' }
 { '7/12'  1 '5/2'  '11/6' }
 { '9/20'  1 '7/2'  '13/3'  '25/12' }
 { '11/30' 1 '9/2'  '47/6'  '77/12'  '137/60' }
 { '13/42' 1 '11/2' '37/3'  '57/4'   '87/10'  '49/20' }
 { '15/56' 1 '13/2' '107/6' '319/12' '459/20' '223/20' '363/140' }

The first three columns are very simple; subsequent columns do not seem obvious to me. Also, neither the numerators nor denominators are in the OEIS. Is there a combinatorial meaning to these numbers?