We form a number triangle by first placing the positive integers along a down-and-right diagonal of an infinite square grid. Then, all other spaces in the triangle are filled by summing the numbers directly up and to the right of that space.

The sequence $1, 3, 8, 20, 48, 112, 256, 576, \ldots$ occurs in the leftmost column of this triangle.

If $S(x) = x + 3x^2 + 8x^3 + 20x^4 + 48x^5 + 112x^6 + 256x^7 + 576x^8 + \ldots$, what is $S(\frac{1}{3})$?