I am trying to find the closed form of the following sequence:
$$ \left\lbrace -\frac{1}{2} , \frac{1}{9} , \frac{13}{100} , \frac{71}{588} , \frac{71}{648} , \frac{1447}{14520} , \frac{617}{6760} , \frac{1061}{12600} , ... \right\rbrace \tag{1}$$
These form the coefficients $a_k$ with $k=1,3,5,...$ of a power series expansion of the following form:
$$f(x) = \sum_{k=1,3,5,...}^\infty a_k x^k = a_1 x + a_3 x^3 + a_5 x^5 + ..., \tag{2}$$
for which I wish to find a closed form. I could not find anything in Wolfram Alpha nor in OEIS. Here is a plot of $(1)$:
Note that these coefficients were found numerically, and the rational forms given above agree up to at least 15 significant digits. If needed I can probably obtain any desired coefficient, although the precision may decrease.
EDIT:
Here is some context, as suggested in the comments. This sequence arises when trying to solve the following integral:
$$I(x) = \frac{1}{2^9\pi^5} \int_{-\infty}^\infty d\tau \left\lbrace \frac{1}{1+\tau^2} \arctan^2 \frac{\tau}{x} + \frac{x}{x^2 + \tau^2} \arctan^2 \tau \right\rbrace \tag{3}$$
with $x>0$. I managed to obtain the following form numerically:
$$I(x) = \frac{1}{2^{13} \pi^4} + \frac{1}{2^{10}\pi^6} \tanh^{-1}(x) \left\lbrace \log(x) - \tanh^{-1}(x) + 2\log 2 \right\rbrace + \frac{1}{2^9 \pi^6} \sum_{k=1,3,5,...} a_k x^k. \tag{4}$$
So finding a closed form for $(1)$ would give me the exact solution of the integral $I(x)$. Note that the integral exhibits the curious symmetry $I(x) = I(1/x)$.
$$\frac{-1}{2k^2}+\frac1{2k}\left(1+\frac12+\cdots+\frac1{(k-1)/2}\right)$$ Where $k$ is the same odd number as in the question