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.
The first few numbers of this kind are:
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:
The pentatope numbers can also be represented as binomial coefficients:
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 (3k2 − k/2)th pentagonal number and the (3k − 1)th pentatope number is always the (3k2 + k/2)th pentagonal number. The (3k)th pentatope number is the generalized pentagonal number obtained by taking the negative index −3k2 + 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.
Pentatope numbers can be represented as the sum of the first n tetrahedral numbers:[2]
and are also related to tetrahedral numbers themselves:
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 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]
In biochemistry, the pentatope numbers represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein.