  Given a commutative ring $R$ with identity, let $H_R$ be the set of sequences
of elements in $R$. We investigate a novel isomorphism between $(H_R, +)$ and
$(\tilde H_R,*)$, where $+$ is the componentwise sum, $*$ is the convolution
product (or Cauchy product) and $\tilde H_R$ the set of sequences starting with
$1_R$. We also define a recursive transform over $H_R$ that, together to the
isomorphism, allows to highlight new relations among some well studied integer
sequences. Moreover, these connections allow to introduce a family of
polynomials connected to the D'Arcais numbers and the Ramanujan tau function.
In this way, we also deduce relations involving the Bell polynomials, the
divisor function and the Ramanujan tau function. Finally, we highlight a
connection between Cauchy and Dirichlet products.
