Consider a function that behaves like this:

F[0,y]={1,1,1,1,1..}

F[1,y]={1,2,3,4,5..}

F[2,y]={1,3,6,10,15..}

F[3,y]={1,4,10,20,35..}

F[4,y]={1,5,15,35,70..}

...

which is simply F[x,y]=Binomial[x+y,x].

Now the challenge is to come up with an equation that explains this behavior:

G[1,y]={0,1,2,3,4,5....}

G[2,y]={0,1,1,3,3,3,6,6,6,6,10,10,10,10,10,15,15,15,15,1 5,15,21...}

G[3,y]={0,1,1,1,4,4,4,4,4,4,10,10,10,10,10,10,10,10,10,1 0,20,20....}

G[4,y]={0,1,1,1,1,5,5,5,5,5,5,5,5,5,5,15...}

...

It shouldn't be too hard to figure out the pattern, but coming up with a function is the tricky part I can't figure out. Anybody want to take a shot at it? This isn't homework.