How many prefix ones?

In the 1920s, the Polish mathematician Jan Lukasiewicz introduced the bracket-free notation for algebraic expressions, known as Polish notation. In prefix (Polish) notation, the operator is placed before its operands. For example, if the infix expression is (b-c/d)/(e*f-(g+h*k)), the prefix form of the segment $c/d$ is $/cd$, and for b-c/d it is $-b/cd$. The segment e*f is $\ast ef$, h*k is $\ast hk$, and g+h*k is +g*hk. Thus, the expression e*f-(g+h*k) becomes -*ef+g*hk in prefix notation, and combining all together for the final division operation, the full prefix expression is /-b/cd-*ef+g*hk.

Given an integer $N$ (1 \leq N \leq 50), determine the number equal to the total number of possible prefix expressions of length $N$ using only the binary operations '+', '-', '*', '/', as well as the required number of unique letters—either the first letters of the Old Georgian alphabet, the last letters of the modern Ukrainian alphabet, or a combination of both with no repetitions. The result should be computed modulo $1000009$, assuming all possible prefix expressions satisfying the above are ordered in descending order according to the value obtained if the operands are taken to be the required amount of consecutive digits from the ninth decimal representation of $e$, starting from the $753$rd digit of the fractional part. To avoid ambiguity, the answer is counted for a fixed set of the required amount of unique letters.