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There is no idea to solve the question for me.

Let $T\subset\mathbb N_{>0}$ be a finite set of positive integers. For each integer $n>0$, define $a_n$ to be the number of all finite sequences $(t_1,t_2,...,t_m)$ with $m\leq n$, $t_i\in T$ for all $i=1,...,m$ and $t_1+...+t_m=n$. Prove that the infinite series $$1+\sum_{n\geq 1}a_nz^n\in\mathbb C[[z]]$$ is a rational function in $z$, and find this rational function.

The definition of series is strange. Can someone help me? Thank you.

gaoxinge
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    Hint: The coefficient at $x^9$ in $(x^1+x^4)(x^1+x^4)(x^1+x^4)$ denotes the number of ways $9$ can be expressed as sum of three integers (whose order matters) where each integer can be equal to $1$ or $4$. – Peter Košinár Feb 17 '14 at 03:36

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