## 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.