Pascal triangle hockey stick and parallelogram features

The "Hockey Stick" attribute indicates that the sum of any diagonals starting from 1 on the outside of the triangle is the number from the last number diagonally in the shape of the hockey stick. When the numbers of Pascal triangles are left-aligned, this means that if you select a number in a Pascal triangle and move a number to the left and add all the numbers in that column to that number, you will get the original number. This sounds very complicated, but it can be explained more clearly by the example in the figure below:

1 1

1 2 from

1

1 3 from

3 from

 1

1 4 from

6 from

 4 1

1 5 from

10 from

 10 5 1

1 6 from

15 from

 20 15 6 1

1 7 from

twenty one 35 from

 35 21 7 1

1 + 3 + 6 + 10 + 15 + 21 = 35

Try these money for yourself to get to know them. This is one of my favorite patterns in the Pascal triangle - it's amazing that this property always works, however, as we're about to see, it's actually not hard to prove!

As an example, I will show the idea behind the proof...

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