Find and .

One should first note that the denominator in the first sum is the denominator of the binomial coefficient , and so

.

Now, the sum in the above result is just the sum of the *n*th row of Pascal’s triangle; since the binomial theorem says that , if we let *x*=1, we get .

Thus,

.

For the second series, each term is the square of that in the first series:

.

Therefore,

.

Now, to find the sum of the squares of these binomial coefficients, we turn to Vandermonde’s convolution, which states that

. (I find the combinatorial proof here to be the simplest and clearest way of establishing this identity.)

Letting *m*=*r*=*n*, we get

,

since (symmetry of Pascal’s triangle). Thus,

,

and so our second series is

.

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Tags: Binomial Coefficient, Binomial Theorem, Math, Monday Math, Pascal's Triangle, Vandermonde's Convolution

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