Summed Fraction Factorials

Algebra Level 5

31!+2!+3!+42!+3!+4!+53!+4!+5!++20011999!+2000!+2001!=abcd!\frac {3}{1! + 2! + 3!} + \frac {4}{2! + 3! + 4!} + \frac {5}{3! + 4! + 5!} + \ldots + \frac {2001}{1999!+ 2000! + 2001!} = \frac {a}{b} - \frac {c}{d!}

The equation above holds true for coprime postive integers aa and bb, and cc and dd. What is the digit sum of (abcd)(abc)(ab)a(abcd)^{(abc)^{(ab)^a}}?

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