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 $\frac{6 \sqrt{3}}{\pi }$: $$\left(\frac{1}{\left(x^2+3\right)^2},\frac{2
   \left(x^2+60\right)}{\left(x^2+12\right)^3},\frac{3 \left(x^4+162
   x^2+9477\right)}{\left(x^2+27\right)^4},\frac{4 \left(x^6+324 x^4+44928
   x^2+2847744\right)}{\left(x^2+48\right)^5},\frac{5 \left(x^8+564 x^6+141750
   x^4+19912500 x^2+1388390625\right)}{\left(x^2+75\right)^6},\frac{6 \left(x^{10}+900
   x^8+366120 x^6+87829920 x^4+13038019200
   x^2+998326798848\right)}{\left(x^2+108\right)^7}, \dots \right)$$

Are you sure about the $564x^6$ term?  All polynomial coeffs are divisible by $k$ with the exception of that one.
