Pascal's Pyramid

Imagine a pyramid made out of blocks, where the first layer has 1 block, the second layer is a \(2\times 2\) square made up of 4 blocks, the third layer is a \(3\times 3\) square made up of 9 blocks, and so on, so that each \(n^\text{th}\) layer is a square made from \(n^2\) blocks.

Each block has a number written on it. The top block has the number 1 written on it, but every other block has the sum of all the blocks that touch its top side written on it. The first five layers are as follows:

What number would be written on the block that is in the \(3^\text{rd}\) column and \(4^\text{th}\) row of the \(100^\text{th}\) layer?

Note: This problem does not require a programming solution, although you may want to use a calculator for the last step!

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