Euler's Number
Problem 330
Published on Sunday, 27th March 2011, 05:00 am; Solved by 315
An infinite sequence of real numbers a(n) is defined for all integers n as follows:
with e = 2.7182818... being Euler's constant.
For example,
a(0) = 

+ 

+ 

+ ... = e − 1 
a(1) = 

+ 

+ 

+ ... = 2e − 3 
a(2) = 

+ 

+ 

+ ... = 

e − 6 
It can be shown that a(n) is of the form 

for integers A(n) and B(n). 
For example a(10) = 

. 
Find A(10^{9}) + B(10^{9}) and give your answer mod 77 777 777.