Number of triangles

What we are dealing with here are pentatope numbers which are the sum of n tetrahedral numbers. The nth tetrahedral number is in turn the sum of the first n triangular numbers.

A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left.

The first few numbers of this kind are :

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 A pentatope with side length 5 contains 70 3-spheres. Each layer represents one of the first five tetrahedral numbers.

The nth number in the above sequence has the formula C(n + 3, 4)

I don't know why this pattern works:

$1~~~5~~~15~~~35~~~70~~~\color{#20A900}{\boxed{126}}$ $~4~~~10~~~20~~~35~~~\boxed{56}~~$ $~~~6~~~10~~~15~~~\boxed{21}$ $~~~4~~~~~5~~~~~\boxed{6}$ $~~~~~1~~~~~\boxed{1}$

(Each lowest number is $1$ and the number above any two is the sum of them.)