I have a problem with derivative of trace of a matrix. Given (symmetric) matrix $A, E$ and $V$, I want to obtain second-order Taylor expansion of $f(A) : = {\tt {tr}}(A^n V)$ near $E$ such that $$ f(A + E) \approx f(A) + \text{vec}(E)^\top \nabla_A f(A) + \frac12 \text{vec}(E)^\top \left(\nabla_A^2 f(A)\right)\text{vec}(E) $$
First derivative is follows: $$ %\frac{\partial {\tt tr}(A^n V)}{\partial A} \nabla_A f(A) = \nabla_A {\tt tr}(A^n V) = \sum_{j=0}^{n-1} A^{j} V A^{n-j-1} $$
However, for the second derivative, it is hard to obtain closed form to me. Any guidance will be helpful. Many thanks!
$$(\mathrm X + h \mathrm M)^n = \mathrm X^n + h \left( \sum_{k=0}^{n-1} \mathrm X^k \mathrm M \mathrm X^{n-1-k} \right) + h^2 \left(\sum_{\begin{array}{c} k_1 + k_2 + k_3 = n-2\\ k_1, k_2, k_3 \geq 0\end{array}} \mathrm X^{k_1} \mathrm M \mathrm X^{k_2} \mathrm M \mathrm X^{k_3} \right) + O (h^3)$$
Hence, the 2nd order term of $\mbox{tr} \left( \mathrm A (\mathrm X + h \mathrm M)^n \right)$ is
$$h^2 \sum_{\begin{array}{c} k_1 + k_2 + k_3 = n-2\\ k_1, k_2, k_3 \geq 0\end{array}} \mbox{tr} \left( \mathrm A \mathrm X^{k_1} \mathrm M \mathrm X^{k_2} \mathrm M \mathrm X^{k_3} \right)$$