I need higher derivatives of the log-partition function $Z(z)=\log \sum_i \exp(z_i)$, has anyone derived the formula?
Looking at concrete values of derivatives up to order 8, evaluated at $z=(1,1,1)$ makes me suspect there's a nice formula in terms of $p_i=\frac{\exp z_i}{Z}$
$$H_2= \left( \begin{array}{ccc} \frac{2}{9} & -\frac{1}{9} & -\frac{1}{9} \\ -\frac{1}{9} & \frac{2}{9} & -\frac{1}{9} \\ -\frac{1}{9} & -\frac{1}{9} & \frac{2}{9} \\ \end{array} \right)=\text{diag}(p)-pp' $$
$$H_3=\left[\left( \begin{array}{ccc} \frac{2}{27} & -\frac{1}{27} & -\frac{1}{27} \\ -\frac{1}{27} & -\frac{1}{27} & \frac{2}{27} \\ -\frac{1}{27} & \frac{2}{27} & -\frac{1}{27} \\ \end{array} \right),\left( \begin{array}{ccc} -\frac{1}{27} & -\frac{1}{27} & \frac{2}{27} \\ -\frac{1}{27} & \frac{2}{27} & -\frac{1}{27} \\ \frac{2}{27} & -\frac{1}{27} & -\frac{1}{27} \\ \end{array} \right),\left( \begin{array}{ccc} -\frac{1}{27} & \frac{2}{27} & -\frac{1}{27} \\ \frac{2}{27} & -\frac{1}{27} & -\frac{1}{27} \\ -\frac{1}{27} & -\frac{1}{27} & \frac{2}{27} \\ \end{array}\right) \right]$$
(crossposted on physics.SE)