On the number of perfectly balanced Boolean functions with barrier of length $3$

Some lower and upper bounds are obtained for the logarithm of the number of Boolean functions with the right barrier of length $3$ essentially depended on the last variable. Also, the following new lower bound for the logarithm of the number of perfectly balanced Boolean functions of $n$ variables with nonlinear dependence on the first and on the last variable is obtained: $2^{n-2}\left(1+\dfrac{\log_25}4-\mathrm O(1/\sqrt n)\right)$.