coefficients of , where we set for simplicity. Suppose then that

Now

The two summations can be reindexed with and combined to yield

Thus the extreme left and right coefficients remain as 1, and for any given , the coefficient of the term in the polynomial is equal to , the sum of the and coefficients in the previous power . This is indeed the downward-addition rule for constructing Pascal's triangle.

It is not difficult to turn this argument into a proof (by

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