I'm trying to solve a problem in two different ways. For the most 'straightforward calculation method' I need that $$\sum_{k=1}^{n+1} \binom{n+1}{k} \sum_{\ell=0}^{k-1} \binom{n}{\ell}=2^{2n}.$$
I looked at some sums involving products of binomial coefficients, like Vandermonde identity but wasn't able to find any applicable one. I also looked to write the partial binomial sums as an integral (which can for example be found on Wikipedia for the cumulative distribution function of the binomial distribution), but this didn't bring me closer.