View source for Pentatope number

{{short description|Number in the 5th cell of any row of Pascal's triangle}}
{{Pascal_triangle_simplex_numbers.svg|2=pentatope numbers}}

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 sphere]]s which can be [[Sphere packing|packed]] into a [[pentatope]] (a 4-dimensional [[tetrahedron]]) of increasing side lengths.

The first few numbers of this kind are:

: [[1 (number)|1]], [[5 (number)|5]], [[15 (number)|15]], [[35 (number)|35]], [[70 (number)|70]], [[126 (number)|126]], [[210 (number)|210]], [[330 (number)|330]], [[495 (number)|495]], [[715 (number)|715]], [[1001 (number)|1001]], [[1365  (number)|1365]] {{OEIS|id=A000332}}

[[Image:Pentatope of 70 spheres animation.gif|frame|right|A [[pentatope]] with side length 5 contains 70 [[3-sphere]]s.  Each layer represents one of the first five [[tetrahedral number]]s.  For example, the bottom (green) layer has 35 [[sphere]]s in total.]]

Pentatope numbers belong to the class of [[figurate number]]s, which can be represented as regular, discrete geometric patterns.<ref>{{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}}</ref>

== 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:

:<math>P_n = \frac{n^{\overline 4}}{4!} = \frac{n(n+1)(n+2)(n+3)}{24} .</math>

The pentatope numbers can also be represented as [[binomial coefficient]]s:

:<math>P_n = \binom{n + 3}{4} ,</math>
which is the number of distinct [[4-tuple|quadruple]]s 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 number]]s. To be precise, the {{math|(3''k'' − 2)}}th pentatope number is always the <math>\left(\tfrac{3k^2 - k}{2}\right)</math>th pentagonal number and the {{math|(3''k'' − 1)}}th pentatope number is always the <math>\left(\tfrac{3k^2 + k}{2}\right)</math>th pentagonal number. The {{math|(3''k'')}}th pentatope number is the [[pentagonal number|generalized pentagonal number]] obtained by taking the negative index <math>-\tfrac{3k^2 + k}{2}</math> in the formula for pentagonal numbers. (These expressions always give [[integer]]s).<ref name="oeis">{{Cite OEIS|A000332}}</ref>

The [[infinite sum]] of the [[Multiplicative inverse|reciprocals]] of all pentatope numbers is {{sfrac|4|3}}.<ref>{{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.&nbsp;435.</ref> This can be derived using [[telescoping series]].

:<math>\sum_{n=1}^\infty \frac{4!}{n(n+1)(n+2)(n+3)} = \frac{4}{3}.</math>

Pentatope numbers can be represented as the sum of the first {{mvar|n}} [[tetrahedral number]]s:<ref name="oeis"/>

:<math>P_n = \sum_{ k =1}^n \mathrm{Te}_k,</math>

and are also related to tetrahedral numbers themselves:

:<math>P_n = \tfrac{1}{4}(n+3) \mathrm{Te}_n.</math>

No [[prime number]] is the predecessor of a pentatope number (it needs to check only -1 and {{nowrap|1=4 = 2<sup>2</sup>}}), and the largest [[semiprime]] which is the predecessor of a pentatope number is 1819.

Similarly, the only primes preceding a [[Figurate_number#Triangular_numbers_and_their_analogs_in_higher_dimensions|6-simplex number]] are [[83 (number)|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 [[quartic equation|Ferrari's method]]:

:<math>n = \frac{\sqrt{5+4\sqrt{24x+1}} - 3}{2}.</math>

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<ref>{{Cite web | url=http://mathworld.wolfram.com/PentatopeNumber.html | title=Wolfram MathWorld site}}</ref>
:<math>\frac{x}{(1-x)^5} = x + 5x^2 + 15x^3 + 35x^4 + \dots .</math>

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

==References==
{{reflist}}

{{Figurate numbers}}
{{Classes of natural numbers}}
[[Category:Figurate numbers]]
[[Category:Simplex numbers]]