It should be clear by now that a database of known hypergeometric closed forms is a useful tool for doing sums of binomial coefficients. We simply convert any given sum into its canonical hypergeometric form, then look it up in the table. If it’s there, fine, we’ve got the answer. If not, we can add it to the database if the sum turns out to be expressible in closed form. We might also include entries in the table that say, “This sum does not have a simple closed form in general.” For example, the sum corresponds to the hypergeometric

this has a simple closed form only if m is near 0, , or n.

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