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Let $F(x,y)=\sum_{n,m\geq0}Q_n^mx^my^m$ be a generating function. Based on the recurrence relation alone, you should be getting $$F(x,y)=\frac{P(x,y)}{1-2y-xy^2};$$ for some polynomial $P(x,y)$ which depends on the initial condition (this, I leave up to you).