I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the differential equation at the end. My work so far is below. Thanks in advance!
$C_N = \frac{\beta}{N(\beta-N)}$
$N(\beta-N)C_N = \beta$
$\sum_{N=1}N(\beta-N)C_N z^N = \sum_{N=1}\beta z^N$
$\sum_{N=1}N\beta C_N z^N -\sum_{N=1}N^2 C_N z^N = \sum_{N=1}\beta z^N$
$\beta zC'(z) - (2\sum_{N=1}$ $N\choose2$ $C_N z^N$ + $\sum_{N=1}N C_Nz^N) = \beta\sum_{N=1} z^N$
$\beta zC'(z) - 2z^2C''(z) - zC'(z) = \beta\sum_{N=1} z^N$
$(\beta - 1)zC'(z) - 2z^2C''(z)=\beta \frac{1}{1-z}$