# Definition:Pentatope Number

## Definition

Pentatope numbers are those denumerating a collection of objects which can be arranged in $4$ dimensions in the form of a regular pentatope.

The $n$th pentatope number $P_n$ is defined as:

$\displaystyle P_n = \sum_{k \mathop = 1}^n T_k$

where $T_k$ is the $k$th tetrahedral number.

### Sequence

The sequence of pentatope numbers begins as follows:

$0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, \ldots$