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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 $h\mapsto +\infty$?

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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.

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  • $\begingroup$ Thank you very much. Is there a simple but fairly good exponential upper bound for $N(h)$ when $h\gg 0$? I found the Hardy-Ramanujan asymptotic partition formula but if I got it right is for $q(n,k)$ when $n\gg k$. In the cases I am considering $n$ and $k$ are related. $\endgroup$
    – Puzzled
    Mar 3, 2023 at 19:14
  • $\begingroup$ @Mor Szekeres's asymptotic formula (summarized in the link in my post) applies for a wide range of pairs $(n,k)$. In particular, it applies for $(n,k)=(6h-6,h-1)$ as $h\to\infty$. On the other hand, the formula is complicated, so you will need some effort to get out of it what you want. $\endgroup$
    – GH from MO
    Mar 3, 2023 at 19:30
  • $\begingroup$ Thank you, I think I almost got it. One last question. What is the role of the $\epsilon > 0$ in Szekeres's theorem? If $\epsilon$ is small then then $e^{n^{1/2}g(u)+O(n^{-1/6+\epsilon})}$ goes as $e^{n^{1/2}g(u)}$ but for $\epsilon \gg 0$ it goes as $e^{n^{-1/6+\epsilon}}$. On the other hand is seems that the formula holds for any $\epsilon > 0$. $\endgroup$
    – Puzzled
    Mar 4, 2023 at 0:13
  • $\begingroup$ @Mor In analysis, $\epsilon$ usually means an arbitrary (small) positive number. I suspect that in your case $n^{1/2} g(u)$ dominates over $n^{-1/2+\epsilon}$ for $\epsilon>0$ sufficiently small, so that the exponent $(1+o(1))n^{1/2} g(u)$ is valid. Note also that $\gg$ does not always mean "much larger than". For example, in analytic number theory, it means that "larger than some constant times of". $\endgroup$
    – GH from MO
    Mar 4, 2023 at 8:48
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$N(h)$ can be expressed via partition function $q$ as $$N(h)=q(6h-6,h-1).$$

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