# 2013-09 Inequality for a sequence

Let $$N > 1000$$ be an integer. Define a sequence $$A_n$$ by
$A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.$
Show that the following inequality holds for any integer $$k$$ with $$1 \leq k \leq (1/2) N^{1/3}$$.
$A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.$

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