poly family

Xitami

[itex]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[/itex]
?

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.