How many ways are there to get from $(0, 0)$ to $(x, y)$ in coordinates system assuming that from $(i, j)$ you can move to $(i + 1, j + 1)$, $(i - 1, j + 1)$ and if you make some moves of the second type in a row then you have to do at least the same amount of moves of the first type in a row. For example, there are $5$ ways to get from $(0, 0)$ to $(-1, 5)$ and $0$ ways to get to $(-2, 5)$. You can assume that $(x, y)$ lies in a second quadrant.
What I've found out is that the problem is equivalent to calculating how many binary sequences of length $y$ are there such that there are $x$ more ones, than zeros and after each contiguous sequence of zeros there must be contiguous sequence of ones of length at least the same. For example, a correct sequence is $1111001110101000111000011111$.