Definition:Pentatope Number
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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:
- $\ds P_n = \sum_{k \mathop = 1}^n T_k$
where $T_k$ is the $k$th tetrahedral number.
Sequence
The sequence of pentatope numbers begins as follows:
- $0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, \ldots$
Also see
- Closed Form for Pentatope Numbers: $P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $56$