  In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal
arrays of integers are constructed that display behaviors found in Pascal's
Triangle. We explain these behaviors further using the binomial transform and
discrete convolution. Here we begin by introducing the notion of a binomial
array and develop several "hockey stick" rules. Then we give an algorithm that
generalizes the classical Vandermonde Identity; this produces infinite families
of summation formulas, which we use to expand and prove certain combinatorial
identities for the Catalan numbers. Finally, we recast the theory in terms of
the finite-dimensional representation theory of $SL(2, F)$.
