Number in the 5th cell of any row of Pascal's triangle
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 in the OEIS)
Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns.
Formula.
The formula for the nth pentatope number is represented by the 4th rising factorial of n divided by the factorial of 4:
formula_1
The pentatope numbers can also be represented as binomial coefficients:
formula_2
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 (3"k" − 2)th pentatope number is always the formula_3th pentagonal number and the (3"k" − 1)th pentatope number is always the formula_4th pentagonal number. The (3"k")th pentatope number is the generalized pentagonal number obtained by taking the negative index formula_5 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.
formula_6
Pentatope numbers can be represented as the sum of the first n tetrahedral numbers:
formula_7
and are also related to tetrahedral numbers themselves:
formula_8
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 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:
formula_9
The number x is pentatope if and only if n is a natural number. In that case x is the nth pentatope number.
Generating function.
The generating function for pentatope numbers is
formula_10
Applications.
In biochemistry, the pentatope numbers represent the number of possible arrangements of "n" different polypeptide subunits in a tetrameric (tetrahedral) protein.
References.
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