  We study the effects of the branching $\mathfrak{osp}(1|2n)\supset
\mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional
$\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$.
In the first part we use combinatorial methods such as Young tableaux and Young
subgroups to construct a new basis for $L(p)$ that respects this branching and
we express the basis elements explicitly in two distinct ways. First as
monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the
$\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in
the generators of $\mathfrak{osp}(1|2n)$ acting on the
$\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. In the second part we
use extremal projectors and the theory of Mickelsson-Zhelobenko algebras to
give new explicit constructions of raising and lowering operators related to
the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$. We use the
raising operators to give new expressions for the elements of the
Gel'fand-Zetlin basis for $L(p)$ as monomials of operators from
$U(\mathfrak{osp}(1|2n))$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight
vector in $L(p)$. We observe that the Gel'fand-Zetlin basis for $L(p)$ is
related to the basis constructed earlier in the paper by a triangular
transition matrix. We end the paper with a detailed example treating the case
$n=3$.
