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Euler's Number

Problem 330

Published on Sunday, 27th March 2011, 05:00 am; Solved by 314
An infinite sequence of real numbers a(n) is defined for all integers n as follows:

For example,

a(0) =
1
1!
 +
1
2!
 +
1
3!
 + ... = e − 1
a(1) =
e − 1
1!
 +
1
2!
 +
1
3!
 + ... = 2e − 3
a(2) =
2e − 3
1!
 +
e − 1
2!
 +
1
3!
 + ... =
7
2
 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 − 652694486
10!
 .

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