Monday Math 134

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 nth 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
.