Ersby's Triangle

Pascal's Triangle is a famous and fascinating piece of mathematical architecture. In its sequences are many interesting patterns that also appear in the mathematics of probability, fractals, and geometry.

One day, I decided to take a closer look at it. In particular, I wanted to find a formula to describe the numbers along each diagonal.

So far, so good. But I came a bit unstuck when I reached the fourth diagonal. At first, I thought I found something to do with the differences between the numbers...

 Eventually, after several false starts I guessed that since the first two numbers in the sequence were 1 and 4 perhaps the answer was something to do with square numbers. Before long, I came up with something else...

The fifth diagonal was also difficult, but when I went back to working only with square numbers, the answer came quite quickly.

I noticed that 1 squared was being multiplied by the following numbers: 1, 1, 2, 2, 3, 3...

This time the multiples of 1 squared were 1, 2, 4, 6, 9... But I was also interested in how the multiples seemed to "move along" the sum because now 2 squared was being multiplied by 1, 1, 2, 2, 3... I wondered if something was going on so, for clarity's sake, I wrote these multiples on a copy of Pascal's triangle, beside the number that each sum belonged to.