= Solution
Choose the prepared <wavefunction> real and positive on its support. Its normalization is
$$
\psi(x)=\frac{\sqrt3}{a^{3/2}}x\quad(0<x<a),\qquad \psi(x)=0\quad\text{elsewhere}.
$$
In the <ramp-state expansion in a symmetric infinite well>, write $\psi=\sum_{n\geq1}c_n\psi_n$. Orthogonality gives
$$
c_n=\frac{\sqrt3}{a^2}\int_0^a x\sin\left(\frac{n\pi(x+a)}{2a}\right)\,dx.
$$
For $\kappa=n\pi/(2a)$, a primitive is $-x\cos(\kappa(x+a))/\kappa+\sin(\kappa(x+a))/\kappa^2$. Evaluating both endpoints gives
$$
\boxed{c_n=-\frac{2\sqrt3(-1)^n}{n\pi}-\frac{4\sqrt3\sin(n\pi/2)}{n^2\pi^2}.}
$$
An equivalent parity-separated form is
$$
\boxed{c_{2p}=-\frac{\sqrt3}{p\pi},\qquad c_{2p-1}=\frac{2\sqrt3}{(2p-1)\pi}\left[1+\frac{2(-1)^p}{(2p-1)\pi}\right].}
$$
These formulas with $\sum_n c_n\psi_n$ are the requested explicit expansion. The convergence is in the square-integrable norm; at the idealized discontinuity at the right wall, a pointwise boundary value does not change the quantum state or its <probability amplitudes>.
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