  Contrary to several other families of lambda terms, no closed formula or
generating function is known and none of the sophisticated techniques devised
in analytic combinatorics can currently help with counting or generating the
set of {\em simply-typed closed lambda terms} of a given size.
  Moreover, their asymptotic scarcity among the set of closed lambda terms
makes counting them via brute force generation and type inference quickly
intractable, with previous published work showing counts for them only up to
size 10.
  By taking advantage of the synergy between logic variables, unification with
occurs check and efficient backtracking in today's Prolog systems, we climb 4
orders of magnitude above previously known counts by deriving progressively
faster Horn Clause programs that generate and/or count the set of closed
simply-typed lambda terms of sizes up to 14. A similar count for {\em closed
simply-typed normal forms} is also derived up to size 14.
  {\em {\bf Keywords:} logic programming transformations, type inference,
combinatorics of lambda terms, simply-typed lambda calculus, simply-typed
normal forms. }
