Pascal Triangle: Can you explain the following result?

Row 0 : 1
Row 1 : 1    1
Row 2 : 1    2    1
Row 3 : 1    3    3    1
Row 4 : 1    4    6    4    1

Etc.


Take a row of Pascal's Triangle.
Write a second copy of the row below it and place 0.
Multiply each pair of numbers and
then sum these products.

For example,

1    1    0
0    1    1
----------------
0    2    0
0 + 2 + 0 = 2 (it is in row 2)

1    2    1    0
0    1    2    1
-------------
0    2    2    0
0 + 2 + 2 + 0 = 4 (in row 4)

1    3    3    1    0
0    1    3    3    1
----------------------
0    3    9    3    0
0 + 3 + 9 + 3 + 0 = 15 (in row 6)

1     4     6     4     1     0
0     1     4     6     4     1
-----------------------------
0    4    24    24    4    0
0 + 4 + 24 + 24 + 4 + 0 = 56 (in row 8)

etc.

Assignment Discovery: Pascal's Triangle

Pascal's Triangle was named after Blaise Pascal in the 1600's. In this triangle, the sum of any two adjacent numbers lies just below them. Learn more with this clip from Discovery Channel's "Assignment Discovery."