Definition 8.6.12
The Bell number is the number of partitions of
.

It is obvious from the definitions that
.

Lemma 8.6.13

Proof.
For, put the element in with a -subset of
for
to .

There isn't a nice closed formula for , but there is a nice expression
for its exponential generating function.

Definition 8.6.14
The exponential generating function that is associated with the sequence
is

If we have
and
(with obvious notation) and
then
, the exponential convolution
of
and
.
Hence is the coefficient of in the exponential
convolution of the sequences
and
.
Thus
. (Shifting is achieved by
differentiation for exponential generating functions.) Therefore
and using the condition
we find
that . So