If A(x)=∑n=0∞n!xnA(x) = \sum_{n=0}^{\infty} n! x^nA(x)=∑n=0∞n!xn, is the recurrence linear?
Yes, of order 1.
Yes, of order 2.
No, it's not a linear homogeneous recurrence with constant coefficients.
Yes, but infinite order.