Plane section recurrences in the Pascal pyramid

Abstract

This work deals with the plane sections lying over the Pascal pyramid. Any plane section can be defined using two main diagonals with parameters $\left(q_1,r_1\right)$ and $\left(q_2,r_2\right)$. We study the main case $r_1=r_2=1$, which corresponds to plane sections crossing all parallel planes to the $x$ axis and intersecting with the integer coordinates of the pyramid. We establish the recurrence relation satisfied by the sequence of numbers counting the sum of elements lying over each plane section.