Answer by Matt for Sum of 'the first k' binomial coefficients for fixed $N$

In particular, does it have a closed form or notable algorithm for computing it efficiently?

Is an $O(k)$ algorithm efficient enough? If so, here is a C++ implementation:

unsigned long long sumbincoef( unsigned N, unsigned k ) {  unsigned long long i, bincoef = 1, sum = 1;  for( i=1 ; i<=k ; ++i ) {    bincoef = bincoef * (N-i+1) / i;    sum += bincoef;  }  return sum;}

Caution: this can overflow for sufficiently large values of $N$ and $k$.

Since one is summing $N\choose i$ for successive $i$, the relevant recursion relation is simply

$${N\choose i} = {N\choose i-1}\frac{N-i+1}{i}$$

so that each term in the sum

$$\sum_{i=0}^k{N\choose i}$$

is calculated from the preceding term in $O(1)$ time.