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),
$$1,2,2,3,3,4,4,4,\dots.$$
$$\beta\_{\ell}(1,2)=\left\lfloor\sqrt{4\ell+1}-1\right\rfloor \qquad
\Longrightarrow \qquad \lim\_{\ell\rightarrow\infty}\frac{\beta^2\_{\ell}(1,2)}{\ell}=4.$$



Limits (growth rates) of power series coefficients
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.


