What is the probability for this Bernoulli trial? So if we have a Bernoulli trial with the property that the accumulated number of successes is twice the accumulated number of failures and the ratio has never exceeded 2. Is the probability of this happening at the $3k$th trial: $$P_{3k}=3^{k-1}(p^2q)^k ?$$

Taking Aaron's interpretation, when $k=3$, i.e. after $9$ steps, there are twelve possible patterns while your expression would suggest there are nine.

The expression should be $$\frac{1}{2k+1}{3k \choose k} (p^2 q)^k$$ related to OEIS A001764.