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How to solve the following recursive equation?

$$a_1=1,a_2=0$$ $$a_{n+2}=n(a_{n+1}-xa_n),\quad n\geqslant 1.\tag{1}$$

From (1) we have $$a_{3}=-x, \quad a_4 = -2 x, $$ $$a_5 = 3 (-2 + x) x, \quad a_6 = 4 x (-6 + 5 x),\cdots$$

This problem showed up when I tried to find convergent series expansion for ${_1F_1(1,9/4+ix,-y)}$ as $y\to+\infty$ and $0

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    You probably don't need more than about 100 terms of this sequence, right? By the 100th term, the degree of the polynomial is 49. Polynomials beyond this degree are probably not very useful. It would be easy to write a computer program to compute the first 100 or so terms. – 2017-02-10
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    Thanks for the suggestion. I can do it with Mathematica. – 2017-02-11

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