pentatope number

{{short description|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 {{nowrap|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 {{OEIS|id=A000332}}

Image:Pentatope of 70 spheres animation.gif with side length 5 contains 70 3-spheres. Each layer represents one of the first five tetrahedral numbers. For example, the bottom (green) layer has 35 spheres in total.]]

Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns.{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=M.|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=162|contribution=3.1 Pentatope numbers and their multidimensional analogues}}

Formula

The formula for the {{mvar|n}}th pentatope number is represented by the 4th rising factorial of {{mvar|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 {{math|n + 3}} objects, and it is read aloud as "{{math|n}} plus three choose four".

Properties

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the {{math|(3k − 2)}}th pentatope number is always the $\left(\tfrac{3k^2 - k}{2}\right)$ th pentagonal number and the {{math|(3k − 1)}}th pentatope number is always the $\left(\tfrac{3k^2 + k}{2}\right)$ th pentagonal number. The {{math|(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).{{Cite OEIS|A000332}}

The infinite sum of the reciprocals of all pentatope numbers is {{sfrac|4|3}}.{{citation|title=Sums of the inverses of binomial coefficients|journal=Fibonacci Quarterly|year=1981|url=http://www.fq.math.ca/Scanned/19-5/rockett.pdf|first=Andrew M.|last=Rockett|volume=19|issue=5|pages=433–437|doi=10.1080/00150517.1981.12430049 }}. Theorem 2, p. 435. 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 {{mvar|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 {{nowrap|1=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 {{mvar|n}}th pentatope number.

Given a positive integer {{mvar|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 {{mvar|x}} is pentatope if and only if {{mvar|n}} is a natural number. In that case {{mvar|x}} is the {{mvar|n}}th pentatope number.

Generating function

The generating function for pentatope numbers is{{Cite web | url=http://mathworld.wolfram.com/PentatopeNumber.html | title=Wolfram MathWorld site}}

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

Applications

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

References

Category:Figurate numbers

Category:Simplex numbers