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

If $0 \leq k \leq N-2$, then\begin{align}\sum_{i = 0}^k \binom{N}{i} = \left\lfloor \frac{(2^N + 1)^N}{2^{N(N-k)}} \right\rfloor \mod (2^N - 1);\end{align}and, if $1 \leq k \leq N$, then\begin{align}\sum_{i = 0}^k \binom{N}{i} = 1 + \left( \left\lfloor \frac{1 - (2^N + 1)^N}{(2^N - 1)\cdot 2^{Nk}} \right\rfloor \mod 2^N \right).\end{align}See Theorem 3.1 and Theorem 3.2 from arXiv:2312.00301.