Pentatope number explained

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, 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:

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

Formula

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

P _{n} = (n ^{\overline})/(4 !) = (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 objects, and it is read aloud as " plus three choose four".

Properties

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the th pentatope number is always the

\left(\tfrac{ 3 k ^{2} - k}{ 2 }\right) th pentagonal number and the th pentatope number is always the \left(\tfrac{ 3 k ^{2} + k}{ 2 }\right) th pentagonal number. The th pentatope number is the generalized pentagonal number obtained by taking the negative index - \tfrac{ 3 k ^{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 .^[1] This can be derived using telescoping series.

\sum ^{inf ty} _{n =1} (4 !)/(n(n +1 )(n +2 )(n +3 )) = (4)/(3) .

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

P _{n} = \sum ^{n} _{k =1} Te _{k,}

and are also related to tetrahedral numbers themselves:

P _{n} = \tfrac{ 1 }{ 4 }(n +3 ) 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 th pentatope number.

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

n = (\sqrt{ 5+4 \sqrt{ 24 x +1)/() } - 3}.

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

Generating function

The generating function for pentatope numbers is^[2]

(x)/(( 1- x) ^{5}) = x + 5 x ^{2} + 15 x ^{3} + 35 x ^{4} + ... .

Applications

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

Notes and References

1. . Theorem 2, p. 435.
2. Web site: Wolfram MathWorld site.