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[tex]a_0 = 1[/tex]

[tex]a_n = \sum_{k=1}^n f(k)\cdot a_{n-k}[/tex]

where f(n) is a known function (in binomial coefficients, powers, and the like).

In general, how would I go about proving that [itex]a_n\sim g(n)[/itex]? I'm working on more closely estimating the function by calculating its value for large n, along with first and second differences. (Suggestions on better methods are welcome, though I think I'm nearly there -- the second differences look suspiciously hyperbolic.)

Any help would be appreciated.