Mysterious polynomial sequence Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $n\in \mathbb{Z}^+$
\begin{align}
P(0)&= 1\\
P(1)&= a\\
P(2)&= a^2+b\\
P(3)&= a^3+2ab\\
P(4)&= a^4+3a^2b+b^2\\
P(5)&= a^5+4a^3b+3ab^2\\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
\end{align}
More steps upon request.
I'll be grateful for any hints!

Hint. Note that the following recurrence holds: for $n\geq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n-k}{k}a^{n-2k}b^k.$$
