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$a_{-i}=0$ for all positive i. We have the recurrence

$$ a_n = \sum_{i=1}^\infty b_ia_{n-m_i} $$

Where $m_i>0$ for all $i$.

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aren't those the catalan numbers? what's $b_i$? – Harry Stern Dec 2 '11 at 6:19
$b_i$ and $m_i$ are two infinite sequences. As far as I know, partition numbers can be expressed in this way. – Chao Xu Dec 2 '11 at 6:26

In signal processing, this is a kind of on IIR filter or in stochastics this is similar to an auto regressive process.

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