  A nonpolynomial one-dimensional quantum potential representing an oscillator,
that can be considered as placed in the middle between the harmonic oscillator
and the isotonic oscillator (harmonic oscillator with a centripetal barrier),
is studied. First the general case, that depends of a parameter $a$, is
considered and then a particular case is studied with great detail. It is
proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$
and the energies $E_n$ of the bound states are explicitly obtained. Finally it
is proven that the solutions determine a family of orthogonal polynomials
${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every
${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii)
They are orthogonal with respect to a new measure obtained by modifying the
classic Hermite measure.
