Let's pass to geometry. The triangular and pyramidal numbers in the Pascal's Triangle bear visual load by their definition.
Let's try to find geometrical parallels in the modified Pascal's Triangle. It will help us to see a spatial model of periodic system of chemical elements.
Figure 11. Periodicity in the structure of electron (on the left) and nuclear (on the right) atom shells. (The motives of illustrations from the Lindner's book  and Britannica are used)
On the left - filling of electron shells. 2, 10, 18, 36, 54, 86, 118 - magic numbers for electron shells (charges of nuclei of ideal gases);
On the right - filling of proton and neutron shells of atom nuclei. 2, 8, 28, 50, 82, 126, 184 - magic numbers for nucleons.
By attentive consideration it is possible to see square
and triangular numbers.