What series is this?

Calculus Level 4

S=83!325!+1287!5129!+\large S=\frac 8{3!} - \frac{32}{5!} + \frac{128}{7!} - \frac{512}{9!} + \cdots

If SS as defined above can be represented as αsinβ \alpha - \sin \beta for some positive integers α\alpha and β\beta , then find α+19β1\alpha + 19 \beta -1.

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