Solving Polynomials of Increasing Degree

[Xitami]

$x^2+2\\\\ \frac{2}{3} x^3 + \frac{13}{3} x\\\\ \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\ \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\ \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\ \dots$
?

 

[DivisionByZro]

What is your question? We can't just guess.

 

[Xitami]

But please guess :-) how to continue?

 

[Muphrid]

How are these polynomials supposed to be related?

 

[micromass]

Apparently, the OP is not a native english speaker and can't describe the problem well.

I guess that he encountered these polynomials while working and he is asking if somebody recognizes them or sees an easy pattern in them.

 

[Xitami]

Thank you Micromass

 

[haruspex]

Clearly the constant term alternates between 0 and 2. At the other end, you can get the leading term from the preceding line by multiplying by 2x/n. So the leading coefficient is 2^n-1/n! So a natural thing to try is:
- multiply each line by n! (starting with n=2 in the first line) to form the poly sequence P_n(x)
- form a new sequence from this according to Q_n(x) = P_n(x) - 2x*P_n-1(x)
The coefficients that result look a little friendlier. The highest prime that occurs in this sample is 19, a lot better than 83.