Problem 330: Euler's Number

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An infinite sequence of real numbers a(n) is defined for all integers n as follows:

For example,a(0) = 11! + 12! + 13! + ... = e − 1 a(1) = e − 11! + 12! + 13! + ... = 2e − 3 a(2) = 2e − 31! + e − 12! + 13! + ... = 72 e − 6

with e = 2.7182818... being Euler's constant.

It can be shown that a(n) is of the form

A(n) e + B(n)n! for integers A(n) and B(n).

For example a(10) =

328161643 e − 65269448610!.

Find A(109) + B(109) and give your answer mod 77 777 777.