11. The generating function of the sequence $\{0,1,2,3, \cdots\}$ is
11. $f(t)=\frac{t}{(1-t)^{2}}$ First, using the conclusion: If the ordinary generating function of the sequence $\left\{a_{0}, a_{1}, a_{2}, \cdots\right\}$ is $A(t)$, then the ordinary generating function of the sequence $\left\{a_{0}, a_{0}+a_{1}, a_{0}+a_{1}+a_{2}, \cdots\right\}$ is $\frac{1}{1-t} A(t)$, we know that the generating function of the sequence $\{1,2,3, \cdots\}$ is $\frac{1}{(1-t)^{2}}=1+2 t+3 t^{2}+4 t^{3}+\cdots$. Since $f(t)=0+t+2 t^{2}+3 t^{3}+\cdots+n t^{n}+\cdots, \frac{f(t)}{t}=1+2 t+3 t^{2}+\cdots$ $+n t^{m-1}+\cdots=\frac{1}{(1-t)^{2}}$, thus $f(t)=\frac{t}{(1-t)^{2}}$.