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