  We propose and test a novel conjectural relation satisfied by the
superconformal index of maximally supersymmetric $U(N)$ gauge theory in four
dimensions. Analogous relations appear to be also valid for the superconformal
indices of a large collection of other gauge theories, as well as for a broad
class of index-like generating functions. The relation expresses the finite $N$
index as a systematic series of corrections to a large $N$ answer. Individual
corrections have an holographic interpretation as the analytic continuation of
contributions from "giant graviton" branes fixed by a specific symmetry
generator.
