Let $N(h)$ be the number of solutions of the following linear diophantine equation: \begin{equation} x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6; \end{equation} where $h\geq 2$ and solution means a vector $(z_1,\dots,z_{h-1})$ of non-negative integers satisfying the equation.
Does there exist a formula for $N(h)$ or at least an explicit expression for the behavior of $N(h)$ for $h\mapsto +\infty$?