Innumber theory, apentatope number (orhypertetrahedral number ortriangulo-triangular number) is a number in the fifth cell of any row ofPascal's triangle starting with the 5-term row1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensionalunit spheres which can bepacked into apentatope (a 4-dimensionaltetrahedron) of increasing side lengths.

The first few numbers of this kind are:

1,5,15,35,70,126,210,330,495,715,1001,1365 (sequenceA000332 in theOEIS)

Pentatope numbers belong to the class offigurate numbers, which can be represented as regular, discrete geometric patterns.^[1]

The formula for thenth pentatope number is represented by the 4thrising factorial ofn divided by thefactorial of 4:

${\displaystyle P_n=\frac{n^{\overline{4}}}{4!}=\frac{n(n+1)(n+2)(n+3)}{24}.}$

The pentatope numbers can also be represented asbinomial coefficients:

${\displaystyle P_n=(\genfrac{}{}{0ex}{}{n+3}{4}),}$

which is the number of distinctquadruples that can be selected fromn + 3 objects, and it is read aloud as "n plus three choose four".

Two of every three pentatope numbers are alsopentagonal numbers. To be precise, the(3k − 2)th pentatope number is always the${\displaystyle \left(\frac{3k^2-k}{2}\right)}$th pentagonal number and the(3k − 1)th pentatope number is always the${\displaystyle \left(\frac{3k^2+k}{2}\right)}$th pentagonal number. The(3k)th pentatope number is thegeneralized pentagonal number obtained by taking the negative index${\displaystyle -\frac{3k^2+k}{2}}$ in the formula for pentagonal numbers. (These expressions always giveintegers).^[2]

Theinfinite sum of thereciprocals of all pentatope numbers is⁠4/3⁠.^[3] This can be derived usingtelescoping series.

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{4!}{n(n+1)(n+2)(n+3)}=\frac{4}{3}.}$

Pentatope numbers can be represented as the sum of the firstntetrahedral numbers:^[2]

${\displaystyle P_n=\sum _{k=1}^n{\mathrm{T}\mathrm{e}}_k,}$

and are also related to tetrahedral numbers themselves:

${\displaystyle P_n=\frac{1}{4}(n+3){\mathrm{T}\mathrm{e}}_n.}$

Noprime number is the predecessor of a pentatope number (it needs to check only −1 and4 = 2²), and the largestsemiprime which is the predecessor of a pentatope number is 1819.

Similarly, the only primes preceding a6-simplex number are83 and 461.

We can derive this test from the formula for thenth pentatope number.

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

${\displaystyle n=\frac{\sqrt{5+4\sqrt{24x+1}}-3}{2}.}$

The numberx is pentatope if and only ifn is anatural number. In that casex is thenth pentatope number.

Thegenerating function for pentatope numbers is^[4]

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

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

• Figurate numbers
• Simplex numbers

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