Pentatope number

A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 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, 715, 1001, 1365 (sequence A000332 in the OEIS)

Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns.^[1] The formula for the nth pentatopic number is:

{n + 3 \choose 4} = \frac{n(n+1)(n+2)(n+3)}{24} = {n^{\overline 4} \over 4!}.

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ((3k² − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k² + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k² + k)/2 in the formula for pentagonal numbers. (These expressions always give integers).^[2]

The infinite sum of the reciprocals of all pentatopal numbers is $4 \over 3$ .^[3] This can be derived using telescoping series.

\sum_{n=1}^\infty {4! \over {n(n+1)(n+2)(n+3)}} = {4 \over 3}

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.^[2]

Test for pentatope numbers

\frac{\sqrt{24n+1}+1}{2}

is triangular number.

then : $8\times {\frac {{\sqrt {24n+1}}+1}{2}}+1$ is perfect square.

References

↑ Deza, Elena; Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers, World Scientific, p. 162, ISBN 9789814355483
1 2 "Sloane's A000332". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ Rockett, Andrew M. (1981), "Sums of the inverses of binomial coefficients" (PDF), Fibonacci Quarterly, 19 (5): 433–437 . Theorem 2, p. 435.

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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