Ponder This Challenge - November 2026 - The SumSum

The SumSum

Given a finite sequence of length $N$, $a_0,a_1, a_2,\ldots,a_{N-1}$, let's first remove the sequence's first element. We then define the sequence of the partial sums of its remaining elements as $a_1, a_1+a_2,a_1+a_2+a_3, \ldots ,\sum_{i=1}^{N-1} a_i$.

For this sequence of length N−1, let's repeat the "remove first element" and "generate new sequence from the partial sums" operations to obtain sequences of length $N-2, N-3, N-4, \ldots,N-(N-1)$ until we end up with a one-element sequence. We call this element The SumSum of the original sequence.

For example, given the 10-element sequence $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, the SumSum of the sequence is 16,796.

Your goal: For $N=10^{10}$ and $p=10^{10}+19$, find the SumSum modulo $p$ of the sequence $a_0,\ldots,a_{N-1}$ defined by $a_i={N \choose i}$.

A Bonus "*" will be given for finding for $N=10^{11}$ and $p=10^{11}+3$ the SumSum modulo $p$ of the sequence given by $a_i = 13^i$.