## Euler's Number

Published on Sunday, 27th March 2011, 05:00 am; Solved by 408;
Difficulty rating: 70%

### Problem 330

An infinite sequence of real numbers a(n) is defined for all integers n as follows: $$a(n) = \begin{cases} 1 & n \lt 0\\ \sum \limits_{i = 1}^{\infty}{\dfrac{a(n - i)}{i!}} & n \ge 0 \end{cases}$$

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.