Ponder This Challenge - November 2026 - The SumSum

The SumSum

Given a finite sequence of length NN, a0,a1,a2,…,aN−1a_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 a1,a1+a2,a1+a2+a3,…,∑i=1N−1aia_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,…,N−(N−1)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 1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10, the SumSum of the sequence is 16,796.

Your goal: For N=1010N=10^{10} and p=1010+19p=10^{10}+19, find the SumSum modulo pp of the sequence a0,…,aN−1a_0,\ldots,a_{N-1} defined by ai=(Ni)a_i={N \choose i}.

A Bonus "*" will be given for finding for N=1011N=10^{11} and p=1011+3p=10^{11}+3 the SumSum modulo pp of the sequence given by ai=13ia_i = 13^i.