= Solution
Let the summands have <mean> mu, <variance> $\sigma^2>0$, and <cumulants> $\kappa_j$. For the standardized sum $Z_n=(S_n-n\mu)/(\sigma\sqrt n)$, expansion of its log-characteristic function gives a normal leading term and successively smaller <cumulant> terms. <Fourier inversion> turns these terms into <Probabilists' Hermite polynomials>. Under sufficient moments and a smooth nonlattice condition, the <Edgeworth expansion> begins
$$
p_{Z_n}(z)=\phi(z)\left[1+\frac{\kappa_3}{6\sigma^3\sqrt n}(z^3-3z)+O(n^{-1})\right]
$$
for bounded z. Higher orders include the fourth <cumulant> and products of lower <cumulants>. This is a central expansion in powers of $n^{-1/2}$; it need not remain positive, normalized, or accurate in far tails after truncation.
For a <saddlepoint density approximation>, require the summand <cumulant-generating function> K to be finite on an interval about zero. At a sum value $s=ny$, find the interior saddle $\widehat t$ satisfying $K'(\widehat t)=y$. <Exponential tilting> makes the tilted <mean> y, so the tilted sum is approximated locally by its <normal probability density> at its <mean>. Undoing the tilt gives
$$
\widehat p_{S_n}(s)=\frac{\exp\{nK(\widehat t)-\widehat t s\}}{\sqrt{2\pi nK''(\widehat t)}}.
$$
Equivalently this follows by deforming the transform-inversion contour through its stationary point. Suitable analytic and nonlattice regularity gives relative error $O(n^{-1})$ at fixed interior y. This approximation incorporates a nonquadratic tail exponent and is positive, while an Edgeworth truncation uses a <normal probability density> times a polynomial. It uses more information from the distribution and stronger exponential-moment assumptions. Its integral may differ from one by a higher-order amount, so positivity does not imply exact normalization.
For the centered <Laplace distribution>, $K(t)=t^2+t^4/2+\cdots$, hence $\kappa_2=2$ and $\kappa_3=0$. Its <skewness> vanishes, so the order-$n^{-1/2}$ Edgeworth correction is zero. With $z=s/\sqrt{2n}$, the requested expansion, omitting terms of order $n^{-1}$ in the standardized correction, is
$$
\boxed{p_{S_n}^{\rm E}(s)=\frac1{\sqrt{4\pi n}}e^{-s^2/(4n)}.}
$$
For bounded z the exact <probability density function> is this expression times $1+O(n^{-1})$. The prefactor is $n^{-1/2}$ because this is the <probability density function> of the unstandardized sum; that change of units should not be confused with the orders inside the <Edgeworth expansion>.
For the saddlepoint, $K'(t)=2t/(1-t^2)$. Write $y=s/n$ and $d=\sqrt{1+y^2}$. The root in $(-1,1)$ is
$$
\widehat t=\frac{y}{1+d},\qquad1-\widehat t^2=\frac2{1+d},\qquad y\widehat t=d-1,\qquad K''(\widehat t)=d(1+d).
$$
These formulas include $s=0$ without a division by y. Substitution yields
$$
\boxed{p_{S_n}^{\rm SP}(s)=\frac{\exp\left\{n\left[\log\frac{1+d}{2}-d+1\right]\right\}}{\sqrt{2\pi n\,d(1+d)}},\qquad d=\sqrt{1+(s/n)^2}.}
$$
This is the leading saddlepoint approximation, without its order-$n^{-1}$ relative correction. Expanding d near one recovers the same central <normal approximation>, while retaining d captures the Laplace-sum large-deviation exponent away from the center.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-42-laplace-approximations.png]
{title=Leading Edgeworth and saddlepoint approximations compared with the exact Laplace-sum density}
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