  We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant
differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove
that these operators lie in the image of the centre of the enveloping algebra
of $\frak{gosp}(1|2n)$. Using this result, we compute explicit formulas for the
eigenvalues of these operators on irreducible summands of
$\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problem
for orthosymplectic Lie superalgebras in the cases that were not addressed in
Sahi-Salmasian-Serganova. Our main technique relies on an explicit calculation
of a certain determinant with polynomial entries.
