An unbiased coin is tossed $n$ times. $H$ is the number of heads and $T$ is the number of tails. What is the expected value of $|H-T|$? In other words, evaluate in closed form:
$$
\frac{1}{2^{n-1}} \sum_{k<n / 2}(n-2 k)\binom{n}{k}
$$
(In this problem, "closed form" means a form that does not involve a series. The given series can be reduced to a single term involving only binomial coefficients, rational functions of $n$ and $2^{n}$, and the largest integer function $[x]$.)
The expected value of $|H-T|$ is $\frac{b}{a}$ when $n=3$. Find the value of $a+b$.

Let's think step by step and output the final answer within \boxed{}.
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