[seqfan] A conjectural relation for Stirling numbers of the 1-st kind

[Vladimir Shevelev]

Dear SeqFans,
 
For Stirling numbers of the 1-st kind, it is known the following formula (Abramowitz and Stegun, the third control relation):
sum{k=m,m+1,...,n}n^(k-m)s(n+1,k+1)=s(n,m),
or, after simple transformations,
sum{i=1,...,n-k}s(n,k+i)(n-1)^i=(n-1)s(n-1,k).
Recently I observed that a close sum 
sum{i=1,...,n-k}C(k+i,k)s(n,k+i)(n-1)^i=0, if n+k is even, and  -2s(n,k), if n+k is odd.  If anyone saw it anywhere or can prove it?
 
Regards,
Vladimir

 Shevelev Vladimir‎