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 (sequence A000332 in the OEIS)

Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns.^[1]

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

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

The pentatope numbers can also be represented as binomial coefficients:

${\displaystyle P_{n}={\binom {n+3}{4}},}$

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".

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

The infinite sum of the reciprocals of all pentatope numbers is ⁠4/3⁠.^[3] This can be derived using telescoping series.

${\displaystyle \sum _{n=1}^{\infty }{\frac {4!}{n(n+1)(n+2)(n+3)}}={\frac {4}{3}}.}$

Pentatope numbers can be represented as the sum of the first n tetrahedral numbers:^[2]

${\displaystyle P_{n}=\sum _{k=1}^{n}\mathrm {Te} _{k},}$

and are also related to tetrahedral numbers themselves:

${\displaystyle P_{n}={\tfrac {1}{4}}(n+3)\mathrm {Te} _{n}.}$

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.

We can derive this test from the formula for the nth 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:

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

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

The generating function for pentatope numbers is^[4]

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

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