Refined Asymptotics and Explicit Recurrences for the numbers of Young tableaux in the (k,l) hook for k+l less than six

This is an etude in experimental semi-rigorous (rigorizable!) mathematics. The leading asymptotics was brilliantly derived by Allan Berele and Amitai Regev for general hooks H(k,l) and general powers z, but what about more refined asymptotics? For small k and l, one can "guess" a linear recurrence (since we live in the holonomic ansatz) and using the Birkhoff-Trjitzinsky method, beautifully implemented in Doron Zeilberger's Maple package AsyRec (that has been incorporated into the present Maple package), we computed amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev asymptotic formula, and especially the impressive constant in front!