Take standard 2d-Heisenberg group over finite ring Z/p. Choose standard generators $x_i, y_i$.
Consider generating polynomial for growth: $ g(t) = \sum_i g_i t^i $ , where $g_i$ are the ball sizes.
What is the explicit expression for g(t) ?
It should be not so complicated and quite well-known. But I cannot Google it.
PS
Heisenberg is central extension of $(Z/p)^2d$ by $(Z/p)$. The expression for g(t) for these groups are clearly quite simple- just take d-th power of the one for $Z/p$. Thus for Heisenberg it should not be much more complicated.
Also we can expect that polynomial should close to d-th power of some smaller polynomial (but can be combination of several polynomials) . To ensure approximate Gaussian distribution for ball sizes.