Sequences full of fractions

Probability Level 3

Let an{a_n} be a sequence of real numbers such that a0=1a_0 = 1 and

ann2+5n+6=an1n2+6n+8+n+1n2+7n+12\frac{a_n}{n^2 + 5n + 6} = \frac{a_{n-1}}{n^2 + 6n + 8} + \frac{n + 1}{n^2 + 7n + 12}

for all integers n1n \geqslant 1. Find a40a_{40}.

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