I am searching for a closed form of the following sequence:
$$\left\lbrace 32,422,3406,22426, 131995, ... \right\rbrace. \tag{1}$$
This shows up in the expansion of a complicated integral. The decomposition in prime numbers is:
$$32 = 2^5, \tag{2.1}$$ $$422 = 2 \cdot 211, \tag{2.2}$$ $$3406 = 2 \cdot 13 \cdot 131, \tag{2.3}$$ $$22426 = 2 \cdot 11213, \tag{2.4}$$ $$131995 = 5 \cdot 26399. \tag{2.5}$$
It is not obvious what to do with that. I am not a specialist of number theory, and I am not aware of techniques that could be used in order to find the closed form. I did search in OEIS.org, without success.
Here is a plot of $(1)$:
A closed form of your sequence could be the following:
$a_n=49865-\frac{1273063}{12}n+\frac{615335}{8}n^2-\frac{278471}{12}n^3+\frac{61071}{24}n^4$
for any $n\in\mathbb{N}\setminus\{0\}$.