I wonder if curve fitting may be useful. I'm using a brute force method, so if there is a better solution please answer!
Let the polynomial be p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f.
We will set up a system of equations now.
2/(2^2-1) = 2/3 = 32a + 16b + 8c + 4d + 2e + f
3/8 = 243a + 81b + 27c + 9d + 3e + f
4/15 = 4^5(a) + 4^4(b) + etc...
You probably see the idea by now. I'm certain this is the tedious method since one wouldn't want to calculate 7^5 by hand, let alone solve a system of equations with monstrous coefficients like these.
Even wolframalpha doesn't accept the equations...