I'm trying to calculate the limit for the sum of binomial coefficients:

$$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$

Numerically it seems to converge rapidly to zero and I have asked at https://math.stackexchange.com/questions/608296/limit-of-sum-i-1n-left-fracn-choose-i2in-sum-j-0i-i-choose but with no replies.

I tried applying Stirling's approximation and also using the fact that ${n \choose i} \leq 2^n/\sqrt{n}$ but that does not work well as the resulting function tends to infinity.

Please accept my apologies if this turns out to have a simple solution.