Solutions of the Harmonic Oscillator are *not* always a Combination of Separable Solutions?

Sometimes the expansions are not obvious. For example The harmonic oscillator time-dependent Schr"odinger equation $$ ipartial_t psi = -frac 12 partial^2_x psi +frac 12 omega^2 x^2 psi $$ has a ``breathing'' solution $$ psi(x,t)= left(frac{omega}{pi}right)^{1/4}frac 1{sqrt{e^{i omega t} +R e^{-iomega t}}}expleft{ - frac omega 2 left(frac{1-R,e^{-2iomega t}}{1+R,e^{-2iomega t}}right)x^2right}, $$ where the parameter $|R|<1$.

Mehler's formula gives expansion in terms of the states as
$$psi(x,t) {=}pi^{1/4}sum_{n=0}^infty e^{-i(n+1/2) omega t} varphi_n(0)(isqrt R)^n frac{varphi_n(sqrt{omega} x)}{(omega)^{1/4}}. $$ Here $$ varphi_n(x)equiv frac{1}{sqrt{2^n n! sqrt{pi}}} H_n(x) e^{-x^2/2} $$ is the normalized $omega=1$ harmonic oscillator wavefunction. Now $varphi_n(0)$ vanishes if $n$ is odd, and $$ pi^{1/4}varphi_{2n}(0)= frac{1}{sqrt{4^n (2n)! } } frac{(2n)!}{n!}(-1)^{n}. $$ so one has found as set of quite "non obvious" expansion coefficients.