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# Problem 330

Euler’s Number

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

For example,
a(0)=$\frac{1}{1!}$+$\frac{1}{2!}$+$\frac{1}{3!}$+…=e-1
a(1)=$\frac{1}{1!}$+$\frac{1}{2!}$+$\frac{1}{3!}$+…=e-1
a(2)=$\frac{2e-3}{1!}$+$\frac{e-1}{2!}$+$\frac{1}{3!}$+…=$\frac{7}{2}$e-6
with e = 2.7182818… being Euler’s constant.

It can be shown that a(n) is of the form $\frac{A(n)e+B(n)}{n!}$ for integers A(n) and B(n).
For example a(10) = $\frac{328161643 e − 652694486}{10!}$.

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

a(0)=$\frac{1}{1!}$+$\frac{1}{2!}$+$\frac{1}{3!}$+…=e-1
a(1)=$\frac{1}{1!}$+$\frac{1}{2!}$+$\frac{1}{3!}$+…=e-1
a(2)=$\frac{2e-3}{1!}$+$\frac{e-1}{2!}$+$\frac{1}{3!}$+…=$\frac{7}{2}$e-6