  A nonnegative real function $f$ is said to be bell-shaped if it converges to
zero at $\pm\infty$ and the $n$th derivative of $f$ changes sign $n$ times for
every $n = 0, 1, 2, \ldots$ In a similar way, we may say that a nonnegative
sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated
difference of $a_k$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$
Bell-shaped functions were recently characterised by Thomas Simon and the first
author. In the present paper we provide an analogous description of bell-shaped
sequences. More precisely, we identify bell-shaped sequences with convolutions
of P\'olya frequency sequences and completely monotone sequences, and we
characterise the corresponding generating functions as exponentials of
appropriate Pick functions.
