pentatope In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...

number is a number in the fifth cell of any row of

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

starting with the 5-term row , either from left to right or from right to left. The first few numbers of this kind are: : 1, 5, 15, 35, 70, 126, 210, 330,

495 __NOTOC__ Year 495 ( CDXCV) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Viator without colleague (or, less frequently, year 1248 ...

, 715, 1001, 1365 Pentatope numbers belong to the class of

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...

s, which can be represented as regular, discrete geometric patterns.

Formula

The formula for the th pentatope number is represented by the 4th

rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...

of divided by the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

of 4: :

P_n = \frac = \frac .

The pentatope numbers can also be represented as

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s: :

P_n = \binom ,

which is the number of distinct

quadruple Quadruple may refer to: * 4-tuple, an ordered list of elements, with four elements * Quad (figure skating), a figure skating jump * Quadruple (computing), a term used as alternative for nibble in some contexts * Quadruple-precision floating-point ...

s 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 number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...

s. To be precise, the th pentatope number is always the th pentagonal number and the th pentatope number is always the th pentagonal number. The th pentatope number is the

generalized pentagonal number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...

obtained by taking the negative index in the formula for pentagonal numbers. (These expressions always give

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s). The

infinite sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

of the reciprocals of all pentatope numbers is . This can be derived using

telescoping series In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after ca ...

. :

\sum_^\infty \frac = \frac.

Pentatope numbers can be represented as the sum of the first

tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...

s: :

P_n = \sum_^n Te_n,

and are also related to tetrahedral numbers themselves: :

P_n = \tfrac(n+3)Te_n.

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

is the predecessor of a pentatope number (it needs to check only -1 and 4=2²), and the largest

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...

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 :

n = \frac.

The number is pentatope if and only if is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

. In that case is the th pentatope number.

Generating function

The

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...

for pentatope numbers is :

\frac = x + 5x^2 + 15x^3 + 35x^4 + \dots .

Applications

biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...

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

References

{{Num-stub Figurate numbers Simplex numbers