The question is whether the below is true.

Sum[Binomial[n, k], {k, 0, s - 1}] == Sum[2^(k - 1)*Binomial[n - k, s - k], {k, 1, s}]

Mathematica can simplify as follows, but it fails to Reduce[] or Solve[].

2^n == Binomial[n, s] Hypergeometric2F1[1, s - n, s + 1, -1] 
+ Binomial[n - 1, s - 1] Hypergeometric2F1[1, 1 - s, 1 - n, 2]