Problem #32
Japanese Robo Awesome-O
Professor Chaos controls the robot Awesome-O by pressing the forward or back button. He presses the forward button n times and the back button n times, one at a time. It returns back to him after he presses the buttons 2n times.
Let a(n,i) be the number of ways such that the number of moves in the forward direction do not exceed the number of moves in the backward direction and the number of positive moves in the forward direction equals the number of positive moves in backward direction exactly for i times.
Given a(3,1)=2, a(3,2)=2 and a(3,3)=1.
Professor Chaos enters an n∗n square board. He can traverse from (0,0) to (n,n) in all possible ways but is restricted to (1,0), (0,1) and (1,1) directions only. Let b(n) be the number of steps on line x=y in all possible paths on which Chaos can travel.
Given b(2) = 6.
Let f(n) = \sum_{i=1}^{n}a(n,i)^{2}
Find \sum_{n=1}^{10}f(n)*b(n)