= Solution
{c}
For uniformly spaced nodes,
$$
\pi_{n,s}(x_s)=h^n\prod_{j\ne s}(s-j)
=(-1)^{n-s}h^ns!(n-s)!.
$$
There are $n-s$ negative factors, coming from $j>s$. Thus the weights fixed by part (b) satisfy this formula exactly. The printed sign $(-1)^s$ agrees for even $n$; for odd $n$ it differs by the common factor $-1$. Rescaling all weights by $(-1)^n$ gives the printed convention without changing the <barycentric interpolation formula>.
For even $n$, the required magnitude is
$$
\left|\frac{w_0}{w_{n/2}}\right|
=\frac{((n/2)!)^2}{n!}
=\binom n{n/2}^{-1}.
$$
<Stirling's formula> gives
$$
\frac{((n/2)!)^2}{n!}
\sim\frac{\pi n(n/(2e))^n}{\sqrt{2\pi n}(n/e)^n}
=\boxed{\frac{\sqrt{\pi n}}{2^{n+1/2}}.}
$$
Thus central weights are exponentially larger in magnitude than endpoint weights. Their alternating signs can cause severe cancellation in polynomial interpolation at equispaced nodes; they reflect the unfavorable high-degree interpolation geometry associated with <Runge's phenomenon>. A barycentric evaluation formula improves the implementation but does not remove the intrinsic sensitivity or guarantee convergence for every smooth function. The comparison with the bounded relative weights at <Chebyshev-Lobatto nodes> is decisive.
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