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 $h\mapsto +\infty$?
Best Answer
As Max Alekseyev observed, $N(h)$ is the number of partitions of $6h-6$ into at most $h-1$ parts. A complicated asymptotic formula exists for this quantity, as a special case of a result of Szekeres (1953). See this paper by Canfield for more detail.