Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $\alpha$'s matrices can be (asymmetric) banded.
Here is an example of such a matrix:
$$\begin{bmatrix} 0 & \alpha_2 & & & \\ \alpha_2 & 0 & \alpha_3 & & & \\ & & \ddots & & \\ & & \alpha_{n-1} & 0 & \alpha_n \\ & & & \alpha_n & 0 \end{bmatrix}$$
I am aware that there are numerical means to do this (more or less accurate), e.g. most notably the software package EXPOKIT.