Q:

How to seek the relation of a sequence of number?

I have the following sequence:
$$
n=2\Rightarrow\frac{1}{8},\frac{6}{8},\frac{1}{8}\\
n=4\Rightarrow\frac{1}{384},\frac{76}{384},\frac{230}{384},\frac{76}{384},\frac{1}{384}\\
n=6\Rightarrow\frac{1}{46080},\frac{722}{23040},\frac{10543}{46080},\frac{5887}{11520},\frac{10543}{46080},\frac{722}{46080},\frac{1}{46080}\\
n=8\Rightarrow\frac{\{1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1\}}{10321920}
$$
I analysis the denominator $n=2, 4, 6, 6, 10, 12,14,16$
$$
8=2^3\\
384=2^7\cdot 3^1\\
46080=2^{10}\cdot 3^2 \cdot 5^1\\
10321920=2^{15}\cdot 3^1 \cdot 5^1 \cdot 7^1\\
3715891200=2^{18}\cdot 3^4 \cdot 5^2 \cdot 7^1\\
1961990553600=2^{15}\cdot 3^5 \cdot 5^2 \cdot 7^1 \cdot 11^1\\
1428329123020800 =2^{25}\cdot 3^5 \cdot 5^2 \cdot 7^2 \cdot 11^1 \cdot 13^1\\
1371195958099968000 = 2^{31}\cdot 3^6 \cdot 5^3 \cdot 7^2 \cdot 11^1 \cdot 13^1
$$
Obviously, they come from the prime
$\{2,3,5,7,11,13\}$
However, for the numerator, it is hard to find the relations

A:

As regards the numerator see here: http://oeis.org/A060187
Given $n$ a row is given by
$$\sum_{i=0}^k(-1)^{k-i}\binom{n+1}{k-i}(2i+1)^n$$
for $k=0,\dots,n$.

