$x^2 y'' + 3xy' + (1 - 2x)y = 0$
I plugged in the guess $y = x^s \sum_{n = 0}^\infty a_n x^n$ to get $(s^2 + 2s + 1)a_0 x^s + \sum_{n = 0}^\infty (n^2 + 2ns + s^2 + 2n + 2s + 1)a_n x^{n + s} - 2a_{n - 1}x^{n + s} = 0$.
The indicial equation indicates that $s = -1$ with Algebraic Multiplicity $2$, and the summation leads to the recurrence relation $a_n = \frac{2}{n^2 + 2ns + s^2 + 2n + 2s + 1} a_{n - 1}$. My understanding is that now I am supposed to solve this recurrence relation and take the partial derivative of the solution with respect to $s$ evaluated (after partial differentiation) at $s = -1$.
So far, I haven't been able to find any really good techniques to solve the recurrence relations produced by power series ODE solution methods; I either write out some terms of the recurrence relation and pray that the pattern doesn't take half an hour to spot, or Wolfram the answer to the recurrence relation. Characteristic equations, generating functions, and $Z$-Transforms have been unhelpful due to the variable coefficients, and although intuition from analogy to ODEs tells me that first-order linear recurrences should be solvable even with variable coefficients, I have yet to find a reliable technique.
Often the pattern matching flash of insight doesn't come to me or takes an extreme amount of trial and error, and now that $s$'s have been introduced into the recurrence relations, the pattern matching approach has become even less productive. Unfortunately, Wolfram's answer to the recurrence relation ($a_0$ set to $\pi$ to prevent lumping) involves Gamma functions, which cannot be differentiated nicely (with or without logarithmic differentiation). I am unsure if the Gamma functions are allowed to be converted into factorials, since in this problem, $s = -1$ and therefore $s + 2$ and $n + s + 2$ are positive integers, but it certainly wouldn't be allowed in problems where $s$ was not an integer.
My suspicion is that Wolfram is treating $s$ as a constant, but I need it to be treating both $n$ and $s$ as variables when solving the recurrence relation, since the plan is to partially differentiate with respect to $s$, and partially differentiating with respect to a constant won't work. However, Wolfram doesn't seem to do multivariable recurrence relations, and I have been once again unable to find techniques.
Am I understanding the prescription of Forbenius solution method correctly, or am I on the wrong track? How can I proceed with recurrence relations like this when the solution isn't obvious from writing out terms?