I'm looking for a closed form for $G(x) = \sum\limits_{t=0}^{\infty} f(t) x^t$ for f(t) of the form $f(t) = t^a$ or more general $f(t) = \sum\limits_{i=0}^{d} a_i t^i$.
I know there are forms for $f(t) = t$ and $f(t) = t^2$ and $f(t) = t^3$.
bg, Johannes
See e.g. Wilf's "generatingfunctionology". For the generating function for the sequence $< p(k) a_k >$, given the generating function $A(z) = \sum_{n \ge 0} a_n z^n$, you get: $$ \sum_{n \ge 0} p(n) a_n z^n = p(z D) A(z) $$ where $D$ is the derivative. I.e., to get the generating function for $< n^2 >$, start with: $$ G(z) = \frac{1}{1 - z} = \sum_{n \ge 0} z^n $$ and then: $$ z \frac{d}{d z} \left( z \frac{d}{d z} \frac{1}{1 - z} \right) = \frac{z + z^2}{(1 - z)^3} $$