Looking for the logic of a sequence from convolution of probability distributions I am trying to detect a pattern in the followin sequence from convolution of a probability distribution (removing the scaling constant $\frac{6 \sqrt{3}}{\pi }$: $$\left(\frac{1}{\left(x^2+3\right)^2},\frac{2
   \left(x^2+60\right)}{\left(x^2+12\right)^3},\frac{3 \left(x^4+162
   x^2+9477\right)}{\left(x^2+27\right)^4},\frac{4 \left(x^6+324 x^4+44928
   x^2+2847744\right)}{\left(x^2+48\right)^5},\frac{5 \left(x^8+564 x^6+141750
   x^4+19912500 x^2+1388390625\right)}{\left(x^2+75\right)^6},\frac{6 \left(x^{10}+900
   x^8+366120 x^6+87829920 x^4+13038019200
   x^2+998326798848\right)}{\left(x^2+108\right)^7}, \dots \right)$$

Given that $\frac{6\sqrt{3}}{\pi}\cdot\frac{1}{(x^2+3)^2}$ is the pdf of a random variable $X$, we have:
$$\varphi_X(t)=\mathbb{E}[e^{itX}]=(1+\sqrt{3}\,|t|)\,e^{-\sqrt{3}\,|t|}$$
hence the characteristic function of $Y=X_1+\ldots+X_n$, where $X_1,\ldots,X_n$ are independent random variables with the same distribution of $X$, is:
$$\varphi_Y(t) = (1+\sqrt{3}|t|)^n\,e^{-n\sqrt{3}\,|t|}$$
and the pdf of $Y$ is given by:
$$f_Y(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}(1+\sqrt{3}\,|t|)^n\,e^{-n\sqrt{3}\,|t|}e^{-itx}\,dt=\frac{1}{\pi}\int_{0}^{+\infty}(1+\sqrt{3}\,t)^n\,e^{-n\sqrt{3}\,t}\cos(tx)\,dt$$
that can be computed through the residue theorem or by integration by parts.
