So recently I came across a math problem, which states the following: You play a game with a fair coin. You start with zero points, and you throw the coin. If you throw heads, you add 1 point to the score; if you get tails, you subtract 1 point (The score can be negative). What is the average amount of throws after which the score will return to zero for the first time?
From there you can proceed to find the probability weighted average of all numbers of throws ((2throws)*(probability of returning to zero after 2 throws) e.t.c.). To find the probability, however, you have to find the number of ways that the score can reach 0 after a certain number of tosses. If the score could reach zero before reaching the 0 again after these many tosses, the number of ways would be given through Pascal´s Triangle; i.e. binomial coefficient. Since the score cannot "cross" the zero line, however, you get a sort of a half of Pascal´s Triangle, with terms at the center line and to the left (WLOG) not being summed into the next rows. Does anyone know, or has an idea about how one could find, the general formula for a given number in the Pascal´s "Half-triangle"?