  A "truncation" of Pascal's triangle is a triangular array of numbers that
satisfies the usual Pascal recurrence but with a boundary condition that
declares some terminal set of numbers along each row of the array to be zero.
Presented here is a family of natural truncations of Pascal's triangle that
generalize a kind of Catalan triangle. The numbers in each array are realized
as differences of binomial coefficients, as counts of certain lattice paths and
tableaux, and as entries of representing matrices for certain linear
transformations of polynomial spaces. Lucas's theorem is applied to determine
precisely those truncations for which the number of odd entries on each row is
a power of two.
