Does this sequence of polynomials have a closed form? Consider the following sequence of polynomials in the variable $d$, which I encountered during a calculation:
$$
1 + 3 d^2
\\
1 + 10 d^2 + 5 d^4
\\
1 + 21 d^2 + 35 d^4 + 7 d^6
\\
1 + 36 d^2 + 126 d^4 + 84 d^6 + 9 d^8
\\
1 + 55 d^2 + 330 d^4 + 462 d^6 + 165 d^8 + 11 d^{10}
\\
1 + 78 d^2 + 715 d^4 + 1716 d^6 + 1287 d^8 + 286 d^{10} + 13 d^{12}
\\
1 + 105 d^2 + 1365 d^4 + 5005 d^6 + 6435 d^8 + 3003 d^{10} + 455 d^{12} + 
 15 d^{14}
\\
1 + 136 d^2 + 2380 d^4 + 12376 d^6 + 24310 d^8 + 19448 d^{10} + 
 6188 d^{12} + 680 d^{14} + 17 d^{16}
\\
...
$$
These polynomials came from a relatively simple calculation, so there might be a simple way to generate them directly. There are some patterns in the prime factorizations of the coefficients, but I can't exactly point out what.
Is there a closed form or something similar for these polynomials, perhaps involving special functions or binomial coefficients?

It looks like $\dfrac{(1+d)^{2k+1}+(1-d)^{2k+1}}{2}$. 
