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.

Also see

 * Closed Form for Pentatope Numbers: $P_n = \dfrac {n \left({n + 1}\right) \left({n + 2}\right) \left({n + 2}\right)} {24}$