How many binary strings with size $2n$ and satisfies the restrictions that 1)Total occurences of 1 equals to that of 0, 2) in any substring that begin with the first character of the string, there are no less occurrences of $1$ than $0$, are possible?
For example, when $n=1$, the only possible string is $(10)$.
When $n=2$, possible strings are $(1100),(1010)$
When $n=3$, possible strings are $(111000),(110100),(110010),(101100),(101010)$
And I did enumerate through all cases for $n=4$, but didn't find a clear pattern...