I presently have computations for $j=1,\ldots,37$ of a family of $2 j$-degree polynomials. From them, it appears that the leading (highest power) coefficients are given (in descending order) by the rules: \begin{equation} d_1=\frac{\left(\frac{17}{2}\right)^n}{\Gamma (n+1)}, \end{equation} \begin{equation} d_2=\frac{2^{-n-2} 17^{n-2} (1109-497 n)}{3 \Gamma (n)}, \end{equation} \begin{equation} d_3=\frac{2^{-n-5} 17^{n-4} (n (n (247009 n-1370262)+3942323)-11308734)}{9 \Gamma (n)}. \end{equation} Also, $d_4$ is the product of \begin{equation} -\frac{2^{-n-7} 17^{n-6} (n-1) n}{405 \Gamma (n+1)} \end{equation} and \begin{equation} 613817365 n^4-5492491130 n^3+30016283027 n^2-173872269670 n+542508998592. \end{equation}
Further, $d_5$ is the product of \begin{equation} \frac{2^{-n-11} 17^{n-8} (n-1) n}{1215 \Gamma (n+1)} \end{equation} and \begin{equation} 305067230405 n^6-4403156498055 n^5+38051293414691 n^4 \end{equation} \begin{equation} -325978342903557 n^3+2137571940201488 n^2-8722204904328012 n+13657232612174832. \end{equation} Continuing, $d_6$ is the product of \begin{equation} -\frac{2^{-n-13} 17^{n-10} (n-2) (n-1) n}{25515 \Gamma (n+1)} \end{equation} and \begin{equation} 212265778915799 n^7-4033760477145378 n^6+46257531538470350 n^5-526319720165886192 n^4 \end{equation} \begin{equation} +5002806671861237555 n^3-35895786322816308558 n^2+169446873953910154824 n-385892347895176978944, \end{equation} while $d_7$ is the product of \begin{equation} \frac{2^{-n-16} 17^{n-12} (n-2) (n-1) n}{1148175 \Gamma (n+1)} \end{equation} and \begin{equation} 527480460605760515 n^9-14061542253335879085 n^8+216128338841103270330 n^7-3070915881213672409050 n^6 \end{equation} \begin{equation} +39074939804872696010811 n^5-414647891239558549971645 n^4+3466800379462987766973880 n^3 \end{equation} \begin{equation} -20874814527662001270399420 n^2+78054176824402526959936464 n-118165465673929410155118720. \end{equation}
I, of course, would hope to eventually fully characterize the family, and may be able to deduce (using, as above, the Mathematica FindSequenceFunction command) one or two more such coefficients. I can test possible $d_8,\ldots$ candidates against the $j=1,\ldots,37$ computations. I have also examined the ratios $r_i=\frac{d_{i+1}}{d_i}$, with the results \begin{equation} {r_1,r_2}= \left\{\frac{(1109-497 n) n}{3468},\frac{n ((1370262-247009 n) n-3942323)+11308734}{6936 (497 n-1109)}\right\}. \end{equation} Actually, the $j$-th $2 j$-degree polynomial, formulas for the leading coefficients of which I have just presented, yields the rule for the coefficient $c_{j+1}$ of the $(4 n-2)$-degree polynomials $p_{n}(k)$ (in monic form [$c_1=1$]--the original leading coefficient having been $\frac{2^{8 n+1}}{(2 n-1)!}$) discussed in App. A of https://www.hindawi.com/journals/amp/2015/621353/, "Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States", by Paul B. Slater and Charles F. Dunkl. (The variable $\alpha$ there has been changed to $n$ here. A Mathematica file containing these five expressions is available upon request [[email protected]].) The sequence of (negative) integer exponents of 2--that is, 0, 2, 5, 7, 11, 13, 16--is found in the OEIS https://oeis.org/search?q=0%2C2%2C5%2C7%2C11&language=english&go=Search .