TITLE: Limits (growth rates) of power series coefficients
QUESTION [6 upvotes]: Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(t)+u_1(t)x+u_2(t)x^2+\cdots$$
where $u_j(t)$ itself is expanded as a polynomial in $t$ (note: coefficients in each $u_j(t)$ are all equal). Now, read-off the coefficients of $G_{m,n}(x,t)$ in the exact order they appear and be listed (including multiplicities) as $\beta_{\ell}(m,n)$.

QUESTION. Is this limit true?
  $$\lim_{\ell\rightarrow\infty}\frac{\beta_{\ell}^2(m,n)}{\ell}=2mn.$$

EXAMPLE 1. If $m=n=1$ then
$$G_{1,1}(x,t)=1+t+(2+2t+2t^2)x+(3+3t+3t^2+3t^3)x^2+\cdots$$
and hence $\beta_{\ell}(1,1)$ starts with (keep in mind: $\beta_1=1, \beta_2=1, \beta_3=2$, etc)
$$1,1,2,2,2,3,3,3,3,\dots.$$
Hence
$$\beta_{\ell}(1,1)=\left\lfloor\frac{\sqrt{8\ell+1}-1}2\right\rfloor \qquad
\Longrightarrow \qquad \lim_{\ell\rightarrow\infty}\frac{\beta^2_{\ell}(1,1)}{\ell}=2.$$
EXAMPLE 2. If $m=1$ and $n=2$ then
$$G_{1,2}(x,t)=1+(2+2t)x+(3+3t)x^2+(4+4t+4t^2)x^3+\cdots$$
and hence $\beta_{\ell}(1,2)$ starts with (keep in mind: $\beta_1=1, \beta_2=2, \beta_3=2$, etc),
$$1,2,2,3,3,4,4,4,\dots.$$
$$\beta_{\ell}(1,2)=\left\lfloor\sqrt{4\ell+1}-1\right\rfloor \qquad
\Longrightarrow \qquad \lim_{\ell\rightarrow\infty}\frac{\beta^2_{\ell}(1,2)}{\ell}=4.$$

REPLY [7 votes]: Yes, the conjectured limit is true. Let $d=\gcd (m,n)$ and $m=m_1d, n=n_1d$. Suppose $a_k$ denotes the number of solutions to $k=m_1r+n_1s$ with $r,s\geq 0$, so that
$$a_0+a_1x+a_2x^2+\cdots =\frac{1}{(1-x^{m_1})(1-x^{n_1})}.$$
We have $\beta_{\ell}(m,n)=Nd$ if and only if $\sum_{i=0}^{N-1}a_i< \ell\le \sum_{i=0}^{N}a_i$.
Notice that $a_i\in \{0,1\}$ for $0\le i\le m_1n_1-1$, and the rest satisfy $a_{mn+i}=1+a_i$.So we have
$$\sum_{i=0}^{N-1}a_i\geq \sum_{i=0}^{m_1n_1\lfloor \frac{N-1}{m_1n_1}\rfloor}a_i\geq m_1n_1\binom{\lfloor \frac{N-1}{m_1n_1}\rfloor}{2}=\frac{N^2}{2m_1n_1}+O(N)$$
and similarly
$$\sum_{i=0}^{N}a_i\le \sum_{i=0}^{m_1n_1\lceil \frac{N}{m_1n_1}\rceil}a_i\le m_1n_1\binom{1+\lceil \frac{N}{m_1n_1}\rceil}{2}=\frac{N^2}{2m_1n_1}+O(N).$$
So we have $$\lim_{\ell\to \infty} \frac{\ell}{\beta_{\ell}^2(m,n)}=\lim_{N\to \infty}\frac{\frac{N^2}{2m_1n_1}}{N^2d^2}=\frac{1}{2mn}$$
as desired.