  Picking binomial coefficients which cannot be divided by a given prime from
Pascal's triangle, we find that they form a set with self-similarity. Essouabri
studied on a class of meromorphic functions associated to the above set. These
functions are related to fractal geometry and it is a problem whether such a
function has a non-real pole on its axis of absolute convergence.
  Essouabri gave a proof of existence of such a non-real pole in the simplest
case. The keys of his proof are Stein's and Wilson's estimates on how fast the
points multiply in Pascal's triangle modulo a prime. This article will give an
extension of Essouabri's result to some cases with certain ways to count the
points in Pascal's triangle modulo a prime which are different from the
traditional one.
