Pentatope number

Pentatope numbers (hypertenrahedral numbers) represent a 4-dimensional generalization of 2-dimensional triangular numbers ; analogous to the 3-dimensional tetrahedral numbers . Due to their relationship to a geometric figure , the pentatope numbers are figured numbers .

The pentatope number is calculated as follows: ${\displaystyle n.}$

${\displaystyle {{n+3} \choose {4}}={\frac {n(n+1)(n+2)(n+3)}{24}}}$

The first pentatop numbers are: (0), 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, ... (sequence A000332 in OEIS )

designation

The name pentatop number is derived from the geometric figure of the pentatope . It is a four-dimensional body that emerges from a three-dimensional tetrahedron. If a pentatope of the same length could be composed of (hyper) spheres, their number would be identical to a pentatope number. ${\displaystyle n}$

Regular figured numbers

The regular figured numbers include:

• Two-dimensional: triangular numbers

The -th triangular number is the sum of the first natural numbers : ${\displaystyle n}$ ${\displaystyle \Delta _{n}}$ ${\displaystyle n}$

${\displaystyle \Delta _{n}=\sum _{k=1}^{n}k=1+2+\cdots +n}$

• Three-dimensional: tetrahedral numbers

The -th tetrahedral number is the sum of the first triangular numbers: ${\displaystyle n}$ ${\displaystyle T_{n}}$ ${\displaystyle n}$

${\displaystyle T_{n}=\sum _{k=1}^{n}\Delta _{k}=\Delta _{1}+\Delta _{2}+\cdots +\Delta _{n}}$

• Four-dimensional: pentatope numbers

The next regular figured numbers are the pentatope numbers, they are created by summing the first tetrahedral numbers : ${\displaystyle n}$

${\displaystyle {\text{Ptop}}_{n}=\sum _{k=1}^{n}T_{k}=T_{1}+\cdots +T_{n}}$

properties

• In the sequence of pentatopes, four numbers are alternating between odd and even.
• All regular figured numbers are in Pascal's triangle , the pentatop numbers are on the fifth diagonal. In particular, the following applies to the -th pentatope number: ${\displaystyle n}$

${\displaystyle {\text{Ptop}}_{n}={{n+3} \choose {4}}}$

• The series of reciprocal values is convergent:

${\displaystyle \sum _{k=1}^{\infty }{\text{Ptop}}_{k}^{-1}=\sum _{k=1}^{\infty }{\frac {24}{k(k+1)(k+2)(k+3)}}={\frac {4}{3}}.}$

• The generating function of the pentatope numbers is:

${\displaystyle {\frac {x}{(1-x)^{5}}}=\mathbf {1} x+\mathbf {5} x^{2}+\mathbf {15} x^{3}+\mathbf {35} x^{4}+\mathbf {70} x^{5}+\cdots }$

Web links

• Jutta Gut's page about figured numbers
• Eric W. Weisstein : Pentatope number . In: MathWorld (English).