## 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.