Generatingfunctionology (1989) [pdf]
math.upenn.edu
math.upenn.edu
Somewhat related, A = B is also a great book:
http://www.math.upenn.edu/~wilf/AeqB.html
It's a very beautiful book that demonstrates how you can prove certain identities such as sin(a+b) = sin(a)cos(b) + cos(a)sin(b) by merely verifying them at a few select values of a and b.
Some reviews:
> after 30 years of extraordinary efforts (largely the efforts of this book's authors), this problem [developing computer programs to simplify hypergeometric sums] is largely solved
And:
> If the complete automation of a major industry within discrete mathematics with relevance to computer science counts as the first miracle, this entertaining accessible exposition by the discoverers themselves counts as the second. ... Seldom do we find such a dramatic mathematical breakthrough placed within the reach of such a large audience so soon.
http://www.math.upenn.edu/~wilf/AeqB.html
As to "an algorithm for evaluating sums of binomial coefficients", some of the obvious kinds of sums have been trivially solvable since well before computers, while the general case of partial sums are still beyond the state of the art, so all in all, I think you're mixing all this up with something else.