Pentatope number

Formula

The formula for the nth 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 . 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 ), 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:

: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.

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.

References

References

1. (2012). "Figurate Numbers". World Scientific.
2. {{Cite OEIS. A000332
3. Rockett, Andrew M.. (1981). "Sums of the inverses of binomial coefficients". Fibonacci Quarterly.
4. "Wolfram MathWorld site".