Pentatope number

In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, 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



Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns.

Formula
The formula for the $n$th pentatope number is represented by the 4th rising factorial of $n$ divided by the factorial of 4:


 * $$P_n = \frac{n^{\overline 4}}{4!} = \frac{n(n+1)(n+2)(n+3)}{24} .$$

The pentatope numbers can also be represented as binomial coefficients:


 * $$P_n = \binom{n + 3}{4} ,$$

which is the number of distinct quadruples that can be selected from $n + 3$ objects, and it is read aloud as "$n$ plus three choose four".

Properties
Two of every three pentatope numbers are also pentagonal numbers. To be precise, the $(3k − 2)$th pentatope number is always the $$\left(\tfrac{3k^2 - k}{2}\right)$$th pentagonal number and the $(3k − 1)$th pentatope number is always the $$\left(\tfrac{3k^2 + k}{2}\right)$$th pentagonal number. The $(3k)$th pentatope number is the generalized pentagonal number obtained by taking the negative index $$-\tfrac{3k^2 + k}{2}$$ in the formula for pentagonal numbers. (These expressions always give integers).

The infinite sum of the reciprocals of all pentatope numbers is $4⁄3$. This can be derived using telescoping series.


 * $$\sum_{n=1}^\infty \frac{4!}{n(n+1)(n+2)(n+3)} = \frac{4}{3}.$$

Pentatope numbers can be represented as the sum of the first $n$ tetrahedral numbers:


 * $$P_n = \sum_{ k =1}^n \mathrm{Te}_k,$$

and are also related to tetrahedral numbers themselves:


 * $$P_n = \tfrac{1}{4}(n+3) \mathrm{Te}_n.$$

No prime number is the predecessor of a pentatope number (it needs to check only -1 and 4 = 22), and the largest semiprime which is the predecessor of a pentatope number is 1819.

Similarly, the only primes preceding a 6-simplex number are 83 and 461.

Test for pentatope numbers
We can derive this test from the formula for the $n$th pentatope number.

Given a positive integer $x$, to test whether it is a pentatope number we can compute the positive root using Ferrari's method:


 * $$n = \frac{\sqrt{5+4\sqrt{24x+1}} - 3}{2}.$$

The number $x$ is pentatope if and only if $n$ is a natural number. In that case $x$ is the $n$th pentatope number.

Generating function
The generating function for pentatope numbers is
 * $$\frac{x}{(1-x)^5} = x + 5x^2 + 15x^3 + 35x^4 + \dots .$$

Applications
In biochemistry, the pentatope numbers represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein.