In number theory, a pentatope number (or hypertetrahedral number or triangulo-triangular number) is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row , either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope (a 4-dimensional tetrahedron) of increasing side lengths.
The first few numbers of this kind are:
: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365
frame|right|A [[pentatope with side length 5 contains 70 3-spheres. Each layer represents one of the first five tetrahedral numbers. For example, the bottom (green) layer has 35 spheres in total.]]
Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns.
Formula
The formula for the th pentatope number is represented by the 4th rising factorial of divided by the factorial of 4:
:<math>P_n = \frac{n^{\overline 4{4!} = \frac{n(n+1)(n+2)(n+3)}{24} .</math>
The pentatope numbers can also be represented as binomial coefficients:
:<math>P_n = \binom{n + 3}{4} ,</math>
which is the number of distinct quadruples that can be selected from objects, and it is read aloud as " plus three choose four".
Properties
Two of every three pentatope numbers are also pentagonal numbers. To be precise, the th pentatope number is always the <math>\left(\tfrac{3k^2 - k}{2}\right)</math>th pentagonal number and the th pentatope number is always the <math>\left(\tfrac{3k^2 + k}{2}\right)</math>th pentagonal number. The th pentatope number is the generalized pentagonal number obtained by taking the negative index <math>-\tfrac{3k^2 + k}{2}</math> in the formula for pentagonal numbers. (These expressions always give integers).
The infinite sum of the reciprocals of all pentatope numbers is . This can be derived using telescoping series.
:<math>\sum_{n=1}^\infty \frac{4!}{n(n+1)(n+2)(n+3)} = \frac{4}{3}.</math>
Pentatope numbers can be represented as the sum of the first tetrahedral numbers:
:<math>\frac{x}{(1-x)^5} = x + 5x^2 + 15x^3 + 35x^4 + \dots .</math>
Applications
In biochemistry, the pentatope numbers represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein.