Can you invert figurate numbers, in general?
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.
Ah, I understand. I haven't tried it but I assume the continuation for other values of x in G[x,y] would be:
x=4: n(n+1)(n+2)(n+3)/24
x=5: n(n+1)(n+2)(n+3)(n+4)/120
...
But then there's the problem of isolating an equation for n for any x. The higher x is, the more difficult it is to solve for n. Is this correct?