[seqfan] A congruence modulo odd prime

[Vladimir Shevelev]

Let p be odd prime. I believe that, if 1<=k<=(p-1)/2, then the k-th derivative in the point x=2: ((x^(p-1-k)-x^k)/(x-1))^(k) |_(x=2)==0 mod p (generally speaking, one cannot replace 2 by another integer).
E.g., if p=7, k=2, then ((x^4-x^2)/(x-1))^(2)=6*x+2=14 in the point x=2.
Another form of the conjecture: sum{j=1,...,p-2*k-1} 2^{j-1}*C(j+k-1, k)==0 (mod p).
Questions: 1) Did anyone see similar congruence? 2) Can anyone prove (disprove) it?

Regards,
Vladimir



 Shevelev Vladimir‎