Flipping a coin (similar to the ballot problem)
I'll tell you where I am so far. Let H(n,k) be the number of good sequences of length n (where heads lead all the time) and k is number of heads-number of tails. You can just define H(n,k)=0 if k>n or k<=0. Ok, so far? Write a recursion relation for H(n+1,k) in terms of H(n,...). That's easy. Now take the whole list [H(n,n),H(n,n-1)...H(n,1)] and figure out how to use it to construct the list [H(n+1,n+1),H(n+1,n),...,H(n+1,1)]. It's pretty easy. You just take the original list, chop off a bit, add some zeros in two different ways and add them. That let's you write a really simple computer program to compute the H(n+1) list from the H(n) list. That's great. Now that you have a simple way to construct the recursion you sit back an wait for your intuition to kick and tell you how to sum the whole thing up. Unfortunately, I'm still waiting. Any good ideas?