Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in the string. (We'll call a binary string with such a property admissible.)
For example: $A(0) = 1$, because the only admissible string is $1.$
Similarly, $A(1) = 1$, because the only admissible string is $011.$
$A(2) = 2,$ because $00111$ and $01011$ are all the admissible strings.
Note that $A(3)$ is at least $5$, since $0001111, 0010111, 0011011, 0100111$ and $0101011$ are among the admissible strings. (Which is to say, this is not the Fibonacci sequence.)
What is a closed formula for $A(k)$? Alternatively, can anyone code up a program that calculates $A(k)$ for small values of $k$?