Ponder This Challenge - November 2026 - The SumSum

The SumSum

Given a finite sequence of length N, a0,a1,a2,…,aN−1, let's first remove the sequence's first element. We then define the sequence of the partial sums of its remaining elements as a1,a1+a2,a1+a2+a3,…,∑i=1N−1ai.

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,…,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=1010 and p=1010+19, find the SumSum modulo p of the sequence a0,…,aN−1 defined by ai=(Ni).

A Bonus "*" will be given for finding for N=1011 and p=1011+3 the SumSum modulo p of the sequence given by ai=13i.