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Regular orthotopic numbers

(Redirected from Measure polytope numbers)

The regular orthotopic numbers are a family of sequences of figurate numbers corresponding to the ${\displaystyle \scriptstyle d\,}$-dimensional regular orthotope for each dimension ${\displaystyle \scriptstyle d\,}$, where ${\displaystyle \scriptstyle d\,}$ is a nonnegative integer. These include the square numbers, the cube numbers and the hypercube numbers for ${\displaystyle \scriptstyle d\,>\,3\,}$.

 d = 0 Regular 0-orthotopic numbers Point numbers Form point (0 (-1)-cell facets) (regular 0-orthotope) d = 1 Regular 1-orthotopic numbers Linear numbers Form segments (2 0-cell facets) (regular 1-orthotope) d = 2 Regular 2-orthotopic numbers Square numbers (tetragonal numbers) Form squares (4 1-cell facets) (regular 2-orthotope) d = 3 Regular 3-orthotopic numbers Cube numbers (hexahedral numbers) Form cubes (6 2-cell facets) (regular 3-orthotope) d = 4 Regular 4-orthotopic numbers Octachoron numbers Form tesseracts (8 3-cell facets) (regular 4-orthotope) d = 5 Regular 5-orthotopic numbers Decateron numbers Form penteracts (10 4-cell facets) (regular 5-orthotope) d = 6 Regular 6-orthotopic numbers Dodecapeton numbers Form hexeracts (12 5-cell facets) (regular 6-orthotope) d = 7 Regular 7-orthotopic numbers Tetradecahexon numbers Form hepteracts (14 6-cell facets) (regular 7-orthotope) d = 8 Regular 8-orthotopic numbers Hexadecahepton numbers Form octeracts (16 7-cell facets) (regular 8-orthotope) ... ... ... ... ... ... d = d Regular d-orthotopic numbers (2d) (d-1)-cell numbers Form (d)-eracts (2d (d-1)-cell facets) (regular d-orthotope)

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The ${\displaystyle \scriptstyle n\,}$th regular ${\displaystyle \scriptstyle d\,}$-orthotopic numbers are given by the formulae[2]

${\displaystyle P_{2^{d}}^{(d)}(n)=n^{d},\,}$

where ${\displaystyle \scriptstyle d\,\geq \,0\,}$ is the dimension and ${\displaystyle \scriptstyle n-1\,}$ is the number of nondegenerate layered regular orthotopes (${\displaystyle \scriptstyle n\,=\,0\,}$ giving no dot and ${\displaystyle \scriptstyle n\,=\,1\,}$ giving a single dot, a degenerate regular orthotope) of the ${\displaystyle \scriptstyle d\,}$-dimensional regular orthotopic number (regular ${\displaystyle \scriptstyle d\,}$-orthotope number.)

Recurrence relation

${\displaystyle P_{2^{d}}^{(d)}(n)=P_{2^{d}}^{(d)}(n-1)+\sum _{i=1}^{d}{(-1)}^{i-1}{\binom {d}{i}}n^{d-i}\,}$

Generating function

${\displaystyle G_{\{n^{d}\}}(x)={\frac {x\ A_{d}(x)}{(1-x)^{d+1}}}\,}$

where

${\displaystyle A_{d}(x)=\sum _{k=1}^{d}A(d,k)\ x^{k-1}\,}$

is the ${\displaystyle \scriptstyle d\,}$th Eulerian polynomial (Cf. Talk:Regular_orthotopic_numbers) whose Eulerian numbers

${\displaystyle A(d,k)|_{k=1}^{d}=\{1,(2^{d}-d-1),...,(2^{d}-d-1),1\}\,}$

can be recursively generated with the triangle of Eulerian numbers.

${\displaystyle G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=G_{\{n^{d}\}}(x)={\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G_{\{n^{i}\}}(x),\,}$ with ${\displaystyle G_{\{n^{0}\}}(x)=G_{\{1\}}(x)={\frac {1}{1-x}}\,}$
${\displaystyle G_{\{P_{2^{d}}^{(d)}(n)\}}(x)={\frac {x(1+(2^{d}-d-1)x+...+(2^{d}-d-1)x^{d-2}+x^{d-1})}{(1-x)^{d+1}}}\,}$
${\displaystyle G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=Li_{-d}(x)\,}$

Method for obtaining the generating functions for successive powers

Since ${\displaystyle \scriptstyle \sum _{i=0}^{\infty }1x^{i}\,=\,{\frac {1}{1-x}},\ |x|\,<\,1\,}$, the generating function of 1 is then[3] [4]

${\displaystyle G_{\{1\}}(x)={\frac {1}{1-x}}.\,}$

Since ${\displaystyle \scriptstyle G_{\{n+1\}}(x)\,=\,{\frac {G_{\{n\}}(x)}{x}}\,=\,G_{\{n\}}(x)+G_{\{1\}}(x)\,}$, the generating function of ${\displaystyle \scriptstyle n\,}$ is then

${\displaystyle G_{\{n\}}(x)=xG_{\{n\}}(x)+x{\frac {1}{1-x}},\,}$ and ${\displaystyle (1-x)G_{\{n\}}(x)={\frac {x}{1-x}},\,}$ which gives
${\displaystyle G_{\{n\}}(x)={\frac {x}{(1-x)^{2}}}.\,}$

Since ${\displaystyle \scriptstyle G_{\{(n+1)^{2}\}}(x)\,=\,{\frac {G_{\{n^{2}\}}(x)}{x}}\,=\,G_{\{n^{2}\}}(x)+2G_{\{n\}}(x)+G_{\{1\}}(x)\,}$, the generating function of ${\displaystyle \scriptstyle n^{2}\,}$ is then

${\displaystyle G_{\{n^{2}\}}(x)=xG_{\{n^{2}\}}(x)+2x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,}$ and ${\displaystyle (1-x)G_{\{n^{2}\}}(x)={\frac {2x^{2}+x(1-x)}{(1-x)^{2}}}={\frac {x(1+x)}{(1-x)^{2}}},\,}$ which gives
${\displaystyle G_{\{n^{2}\}}(x)={\frac {x(1+x)}{(1-x)^{3}}}.\,}$

Since ${\displaystyle \scriptstyle G_{\{(n+1)^{3}\}}(x)\,=\,{\frac {G_{\{n^{3}\}}(x)}{x}}\,=\,G_{\{n^{3}\}}(x)+3G_{\{n^{2}\}}(x)+3G_{\{n\}}(x)+G_{\{1\}}(x)\,}$, the generating function of ${\displaystyle \scriptstyle n^{3}\,}$ is then

${\displaystyle G_{\{n^{3}\}}(x)=xG_{\{n^{3}\}}(x)+3x{\frac {x(1+x)}{(1-x)^{3}}}+3x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,}$ and ${\displaystyle (1-x)G_{\{n^{3}\}}(x)={\frac {3x^{2}(1+x)+3x^{2}(1-x)+x(1-x)^{2}}{(1-x)^{3}}}={\frac {x(1+4x+x^{2})}{(1-x)^{3}}},\,}$ which gives
${\displaystyle G_{\{n^{3}\}}(x)={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}.\,}$

Since ${\displaystyle \scriptstyle G_{\{(n+1)^{4}\}}(x)\,=\,{\frac {G_{\{n^{4}\}}(x)}{x}}\,=\,G_{\{n^{4}\}}(x)+4G_{\{n^{3}\}}(x)+6G_{\{n^{2}\}}(x)+4G_{\{n\}}(x)+G_{\{1\}}(x)\,}$, the generating function of ${\displaystyle \scriptstyle n^{4}\,}$ is then

${\displaystyle G_{\{n^{4}\}}(x)=xG_{\{n^{4}\}}(x)+4x{\frac {x(1+4x+x^{2})}{(1-x)^{4}}}+6x{\frac {x(1+x)}{(1-x)^{3}}}+4x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,}$ and ${\displaystyle (1-x)G_{\{n^{4}\}}(x)={\frac {4x^{2}(1+4x+x^{2})+6x^{2}(1+x)(1-x)+4x^{2}(1-x)^{2}+x(1-x)^{3}}{(1-x)^{4}}}={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{4}}},\,}$ which gives
${\displaystyle G_{\{n^{4}\}}(x)={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{5}}}.\,}$

Since ${\displaystyle \scriptstyle G_{\{(n+1)^{5}\}}(x)\,=\,{\frac {G_{\{n^{5}\}}(x)}{x}}\,=\,G_{\{n^{5}\}}(x)+5G_{\{n^{4}\}}(x)+10G_{\{n^{3}\}}(x)+10G_{\{n^{2}\}}(x)+5G_{\{n\}}(x)+G_{\{1\}}(x)\,}$, the generating function of ${\displaystyle \scriptstyle n^{5}\,}$ is then

${\displaystyle G_{\{n^{5}\}}(x)=xG_{\{n^{5}\}}(x)+...,\,}$ and ${\displaystyle (1-x)G_{\{n^{5}\}}(x)=...={\frac {x(1+26x+66x^{2}+26x^{3}+x^{4})}{(1-x)^{5}}},\,}$ which gives
${\displaystyle G_{\{n^{5}\}}(x)={\frac {x(1+26x+66x^{2}+26x^{3}+x^{4})}{(1-x)^{6}}}.\,}$

And the general recurrence equation for the generating function of powers is

${\displaystyle G_{\{n^{d}\}}(x)={\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G_{\{n^{i}\}}(x),\,}$ with ${\displaystyle G_{\{n^{0}\}}(x)=G_{\{1\}}(x)={\frac {1}{1-x}}\,}$

Simpler method for obtaining the generating functions for successive powers

Since ${\displaystyle \scriptstyle \sum _{i=0}^{\infty }1x^{i}\,=\,{\frac {1}{1-x}},\ |x|\,<\,1\,}$, we have

${\displaystyle G_{\{n^{0}\}}(x)={\frac {1}{1-x}},\,}$

then

${\displaystyle G_{\{n^{d}\}}(x)=x\ {G'}_{\{n^{d-1}\}}(x)=x\ {\frac {dG_{\{n^{d-1}\}}(x)}{dx}},\ d\geq 1.\,}$ [5]

Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and ${\displaystyle \scriptstyle k\,}$ ${\displaystyle \scriptstyle k\,}$-gonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[6] Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of ${\displaystyle \scriptstyle k\,}$ ${\displaystyle \scriptstyle k\,}$-gonal numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)[7]

A nonempty subset ${\displaystyle \scriptstyle A\,}$ of nonnegative integers is called a basis of order ${\displaystyle \scriptstyle g\,}$ if ${\displaystyle \scriptstyle g\,}$ is the minimum number with the property that every nonnegative integer can be written as a sum of ${\displaystyle \scriptstyle g\,}$ elements in ${\displaystyle \scriptstyle A\,}$. Lagrange’s sum of four squares can be restated as the set ${\displaystyle \scriptstyle \{n^{2}\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,}$ of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every ${\displaystyle \scriptstyle k\,\geq \,3}$, the set ${\displaystyle \scriptstyle \{P(k,n)\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,}$ of ${\displaystyle \scriptstyle k\,}$-gonal numbers forms a basis of order ${\displaystyle \scriptstyle k\,}$, i.e. every nonnegative integer can be written as a sum of ${\displaystyle \scriptstyle k\,}$ ${\displaystyle \scriptstyle k\,}$-gonal numbers.

We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number ${\displaystyle \scriptstyle g(d)\,}$ such that every nonnegative integer is a sum of ${\displaystyle \scriptstyle g(d)\,}$ ${\displaystyle \scriptstyle d\,}$th powers, i.e. the set ${\displaystyle \scriptstyle \{n^{d}\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,}$ of ${\displaystyle \scriptstyle d\,}$th powers forms a basis of order ${\displaystyle \scriptstyle g(d)\,}$. The Hilbert-Waring problem is concerned with the study of ${\displaystyle \scriptstyle g(d)\,}$ for ${\displaystyle \scriptstyle d\,\geq \,2\,}$. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

The (presumed) solution to Waring's problem is (cf. A002804)

${\displaystyle g(n)=(2^{n}-2)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor }=2(2^{n-1}-1)+{\bigg \lfloor }{{{\bigg (}{\frac {3}{2}}{\bigg )}}^{n}}{\bigg \rfloor },\,}$

with series representation

${\displaystyle g(n)=-{5 \over 2}+{\bigg (}{3 \over 2}{\bigg )}^{n}+2^{n}+{\sum _{k=1}^{\infty }{\sin(({3 \over 2})^{n}2k\pi ) \over k} \over \pi }\,}$ for ${\displaystyle ({(3/2)}^{n}\in \mathbb {R} \,}$ and ${\displaystyle ({3/2)}^{n}\not \in \mathbb {Z} )\,}$.

(Presumed) solution to Waring's problem
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
${\displaystyle \scriptstyle (2^{n}-2)+\lfloor {{({3 \over 2})}^{n}}\rfloor \,}$ 1 4 9 19 37 73 143 279 548 1079 2132 4223 8384 16673 33203

Differences

${\displaystyle P_{2^{d}}^{(d)}(n)-P_{2^{d}}^{(d)}(n-1)=\sum _{k=0}^{d-1}{\binom {d}{k}}(n-1)^{k}\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}P_{2^{d}}^{(d)}(n)=H_{m}^{(-d)}\,}$ [8]

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{\frac {1}{P_{2^{d}}^{(d)}(n)}}=H_{m}^{(d)}\,}$ [8]

Sum of reciprocals

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{2^{d}}^{(d)}(n)}}=\zeta (d)\,}$ [9]

Number of j-dimensional "vertices"

${\displaystyle N_{j}=2^{d-j}{\binom {d}{j}},\ (0\leq j\leq d)\,}$

Table of formulae and values

${\displaystyle \scriptstyle N_{0},\,N_{1},\,N_{2},\,N_{3},\,\ldots \,}$ are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the (${\displaystyle \scriptstyle d-1\,}$)-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number ${\displaystyle \scriptstyle N_{0}\,}$ of vertices.

Regular orthotopic numbers formulae and values
d Name

Regular

${\displaystyle \scriptstyle d\,}$-orthotope

${\displaystyle 2d\ (d-1)\,}$-cell

(${\displaystyle N_{0},\,N_{1},\,N_{2},\,\ldots \,}$)

Schläfli symbol[10]

Formulae

${\displaystyle P_{2^{d}}^{(d)}(n)=\,}$

${\displaystyle n^{d}\,}$

${\displaystyle n\,}$ = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

0 Point numbers

0-orthotope

zero-(-1)-cell

()

{}

${\displaystyle n^{0},\ n\geq 1,\,}$

${\displaystyle 0,\ n=0.\,}$

0 1 1 1 1 1 1 1 1 1 1 1 1 (NOT A057427) [11]
1 Segment numbers

1-orthotope

di-0-cell

(2)

{}

${\displaystyle n\,}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 A001477
2 Square numbers

2-orthotope

tetra-1-cell

(4, 4)

{4}

${\displaystyle n^{2}\,}$ 0 1 4 9 16 25 36 49 64 81 100 121 144 A000290
3 Cube numbers

3-orthotope

hexa-2-cell

(8, 12, 6)

{4, 3}

${\displaystyle n^{3}\,}$ 0 1 8 27 64 125 216 343 512 729 1000 1331 1728 A000578
4 Tesseract numbers

4-orthotope

octa-3-cell

(16, 32, 24, 8)

{4, 3, 3}

${\displaystyle n^{4}\,}$ 0 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 A000583
5 Penteract numbers

5-orthotope

deca-4-cell

(32, 80, 80, 40, 10)

{4, 3, 3, 3}

${\displaystyle n^{5}\,}$ 0 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 248832 A000584
6 Hexeract numbers

6-orthotope

dodeca-5-cell

(64, 192, 240, 160, 60, 12)

{4, 3, 3, 3, 3}

${\displaystyle n^{6}\,}$ 0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 A001014
7 Hepteract numbers

7-orthotope

(128, 448, 672, 560, 280, 84, 14)

{4, 3, 3, 3, 3, 3}

${\displaystyle n^{7}\,}$ 0 1 128 2187 16384 78125 279936 823543 2097152 4782969 10000000 19487171 35831808 A001015
8 Octeract numbers

8-orthotope

(256, 1024, 1792, 1792, 1120, 448, 112, 16)

{4, 3, 3, 3, 3, 3, 3}

${\displaystyle n^{8}\,}$ 0 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000 214358881 429981696 A001016
9 Enneract numbers

9-orthotope

(512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18)

{4, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle n^{9}\,}$ 0 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 2357947691 5159780352 A001017
10 Dekeract numbers

10-orthotope

icosa-9-cell

(1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20)

{4, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle n^{10}\,}$ 0 1 1024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401 10000000000 25937424601 61917364224 A008454
11 Hendekeract numbers

11-orthotope

icosidi-10-cell

(..., 22)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle n^{11}\,}$ 0 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609 100000000000 285311670611 743008370688 A008455
12 Dodekeract numbers

12-orthotope

icositetra-11-cell

(..., 24)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle n^{12}\,}$ 0 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256 A008456

Table of related formulae and values

${\displaystyle \scriptstyle N_{0},\,N_{1},\,N_{2},\,N_{3},\,\ldots \,}$ are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the (${\displaystyle \scriptstyle d-1\,}$)-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number ${\displaystyle \scriptstyle N_{0}\,}$ of vertices.

Regular orthotopic numbers related formulae and values
${\displaystyle d\,}$ Name

Regular ${\displaystyle \scriptstyle d\,}$-orthotope

${\displaystyle \scriptstyle 2d\ (d-1)\,}$-cell

${\displaystyle \scriptstyle (N_{0},\,N_{1},\,N_{2},\,\ldots )\,}$

Generating

function

${\displaystyle G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=\,}$

${\displaystyle {\frac {x\ A_{d}(x)}{(1-x)^{n+1}}},\,}$
where ${\displaystyle \scriptstyle A_{d}(x)\,}$ is the ${\displaystyle \scriptstyle d\,}$th Eulerian polynomial
(Cf. Talk:Regular_orthotopic_numbers)
given by ${\displaystyle \scriptstyle A_{d}(x)\,=\,\sum _{k=1}^{d}A(d,k)\ x^{k-1},\,}$
where ${\displaystyle \scriptstyle A(d,k)|_{k=1}^{d}\,=\,\{1,(2^{d}-d-1),\dots ,(2^{d}-d-1),1\}\,}$
are the Eulerian numbers.
(Cf. triangle of Eulerian numbers)

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

Order

of basis

${\displaystyle g_{\{P_{2^{d}}^{(d)}\}}\,}$

Differences[12]

${\displaystyle P_{2^{d}}^{(d)}(n)-\,}$

${\displaystyle P_{2^{d}}^{(d)}(n-1)=\,}$

${\displaystyle \sum _{k=0}^{d-1}{\binom {d}{k}}(n-1)^{k}\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}{P_{2^{d}}^{(d)}(n)}=\,}$

${\displaystyle H_{m}^{(-d)}\,}$ [8][13]

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{1 \over {P_{2^{d}}^{(d)}(n)}}=}$

${\displaystyle H_{m}^{(d)}\,}$ [8][13]

Sum of reciprocals[14]

${\displaystyle \sum _{n=1}^{\infty }{1 \over {P_{2^{d}}^{(d)}(n)}}=}$

${\displaystyle \zeta (d),\,}$[9]

${\displaystyle \zeta (2k)=\,}$

${\displaystyle {{2^{2k-1}|B_{2k}|\pi ^{2k}} \over {(2k)!}},\,}$

${\displaystyle (k\in \mathbb {N} ^{+})\,}$[15]

0 Point numbers

0-orthotope

zero-(-1)-cell

()

{}

${\displaystyle {\frac {x\ A_{0}(x)}{(1-x)}},\,}$

with ${\displaystyle \scriptstyle A_{0}(x)=1\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{-1}{\binom {1}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle \infty \,}$ ${\displaystyle 0,\ n\neq 1,\,}$

${\displaystyle 1,\ n=1.\,}$

${\displaystyle H_{m}^{(0)}\,}$

${\displaystyle m\,}$

${\displaystyle H_{m}^{(0)}\,}$

${\displaystyle m\,}$

${\displaystyle \scriptstyle \lim _{m\to \infty }H_{m}^{(0)}\,}$

${\displaystyle \scriptstyle \,\sim \,m\,\to \,\infty \,}$

1 Segment numbers

1-orthotope

di-0-cell

(2)

{}

${\displaystyle {\frac {x\ A_{1}(x)}{(1-x)^{2}}},\,}$

with ${\displaystyle \scriptstyle A_{1}(x)\,=\,1\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{0}{\binom {1}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle P_{3}^{(2)}(m)=t_{m}}$[2]

${\displaystyle \scriptstyle {\binom {m+1}{2}}\,=\,{{m^{(2)}} \over 2!}\,=\,\left(\!\!{\binom {m}{2}}\!\!\right)\,}$ [16][17]

${\displaystyle H_{m}^{(-1)}\,}$

${\displaystyle H_{m}=H_{m}^{(1)}\,}$ ${\displaystyle \scriptstyle \lim _{m\to \infty }H_{m}\,}$

${\displaystyle \scriptstyle \,\sim \,log(m)\,\to \,\infty \,}$

2 Square numbers

2-orthotope

tetra-1-cell

(4, 4)

{4}

${\displaystyle {\frac {x\ A_{2}(x)}{(1-x)^{3}}},\,}$

with ${\displaystyle \scriptstyle A_{2}(x)\,=\,1+x\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{1}{\binom {2}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 4\,}$ ${\displaystyle 2n-1\,}$ ${\displaystyle {1 \over 4}{\binom {2m+2}{3}}\,}$

${\displaystyle {\frac {(2m)^{(3)}}{4!}}\,}$

${\displaystyle H_{m}^{(-2)}\,}$

${\displaystyle H_{m}^{(2)}\,}$ ${\displaystyle \zeta (2)={{\pi ^{2}} \over 6}\,}$

Base 10: A013661
CFrac: A013679

3 Cube numbers

3-orthotope

hexa-2-cell

(8, 12, 6)

{4, 3}

${\displaystyle {\frac {x\ A_{3}(x)}{(1-x)^{4}}}\,}$

with ${\displaystyle \scriptstyle A_{3}(x)=1+4x+x^{2}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{2}{\binom {3}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 9\,}$ ${\displaystyle 3(n-1)^{2}+3(n-1)+1\,}$

Hex (or centered hexagonal)

numbers[19]

${\displaystyle T_{m}^{2}\,}$

${\displaystyle H_{m}^{(-3)}\,}$

${\displaystyle H_{m}^{(3)}\,}$ ${\displaystyle \zeta (3)\,}$[20]

Base 10: A002117
CFrac: A013631

4 Tesseract numbers

4-orthotope

octa-3-cell

(16, 32, 24, 8)

{4, 3, 3}

${\displaystyle {\frac {x\ A_{4}(x)}{(1-x)^{5}}}\,}$

with ${\displaystyle \scriptstyle A_{4}(x)\,=\,1+11x+11x^{2}+x^{3}\,}$

or ${\displaystyle \scriptstyle A_{4}(x)\,=\,(1+x)(1+10x+x^{2})\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{3}{\binom {4}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 19\,}$ ${\displaystyle \scriptstyle 4(n-1)^{3}+6(n-1)^{2}+4(n-1)+1\,}$

Rhombic dodecahedral

numbers [21]

${\displaystyle \scriptstyle {{(2m)^{(3)}} \over 5!}(6T_{m}-1)\,}$

${\displaystyle H_{m}^{(-4)}\,}$

${\displaystyle H_{m}^{(4)}\,}$ ${\displaystyle \zeta (4)={{\pi ^{4}} \over 90}\,}$

Base 10: A013662
CFrac: A013680

5 Penteract numbers

5-orthotope

deca-4-cell

(32, 80, 80, 40, 10)

{4, 3, 3, 3}

${\displaystyle {\frac {x\ A_{5}(x)}{(1-x)^{6}}}\,}$

with ${\displaystyle \scriptstyle A_{5}(x)\,=\,1+26x+66x^{2}+26x^{3}+x^{4}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{4}{\binom {5}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 37\,}$ ${\displaystyle \scriptstyle 5(n-1)^{4}+10(n-1)^{3}+10(n-1)^{2}+5(n-1)+1\,}$ ${\displaystyle \scriptstyle {{T_{m}^{2}} \over 3}(2m^{2}+2m-1)\,}$

${\displaystyle H_{m}^{(-5)}\,}$

${\displaystyle H_{m}^{(5)}\,}$ ${\displaystyle \zeta (5)\,}$

Base 10: A013663
CFrac: A013681

6 Hexeract numbers

6-orthotope

dodeca-5-cell

(64, 192, 240, 160, 60, 12)

{4, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{6}(x)}{(1-x)^{7}}}\,}$

with ${\displaystyle \scriptstyle A_{6}(x)\,=\,1+57x+302x^{2}+302x^{3}+57x^{4}+x^{5}\,}$

or ${\displaystyle \scriptstyle A_{6}(x)\,=\,(1+x)(1+56x+246x^{2}+56x^{3}+x^{4})\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{5}{\binom {6}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 73\,}$ ${\displaystyle \sum _{i=1}^{6}{\binom {6}{i}}(n-1)^{6-i}\,}$ ${\displaystyle H_{m}^{(-6)}\,}$ ${\displaystyle H_{m}^{(6)}\,}$ ${\displaystyle \zeta (6)={{\pi ^{6}} \over 945}\,}$

Base 10: A013664
CFrac: A013682

7 Hepteract numbers

7-orthotope

(128, 448, 672, 560, 280 , 84, 14)

{4, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{7}(x)}{(1-x)^{8}}},\,}$

with ${\displaystyle \scriptstyle A_{7}(x)\,=\,\,}$

${\displaystyle \scriptstyle 1+120x+1191x^{2}+2416x^{3}+1191x^{4}+120x^{5}+x^{6}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{6}{\binom {7}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 143\,}$ ${\displaystyle \sum _{i=1}^{7}{\binom {7}{i}}(n-1)^{7-i}\,}$ ${\displaystyle H_{m}^{(-7)}\,}$ ${\displaystyle H_{m}^{(7)}\,}$ ${\displaystyle \zeta (7)\,}$

Base 10: A013665
CFrac: A013683

8 Octeract numbers

8-orthotope

(256, 1024 , 1792, 1792, 1120, 448, 112, 16)

{4, 3, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{8}(x)}{(1-x)^{9}}},\,}$

with ${\displaystyle \scriptstyle A_{8}(x)\,=\,\sum _{k=1}^{8}A(8,k)\ x^{k-1},\,}$

where ${\displaystyle \scriptstyle A(8,k)|_{k=1}^{8}\,=\,\,}$

${\displaystyle \scriptstyle \{1,247,4293,15619,15619,4293,247,1\}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{7}{\binom {8}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 279\,}$ ${\displaystyle \sum _{i=1}^{8}{\binom {8}{i}}(n-1)^{8-i}\,}$ ${\displaystyle H_{m}^{(-8)}\,}$ ${\displaystyle H_{m}^{(8)}\,}$ ${\displaystyle \zeta (8)={{\pi ^{8}} \over 9450}\,}$

Base 10: A013666
CFrac: A013684

9 Enneract numbers

9-orthotope

(..., 18)

{4, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{9}(x)}{(1-x)^{10}}}\,}$

with ${\displaystyle \scriptstyle A_{9}(x)\,=\,\sum _{k=1}^{9}A(9,k)\ x^{k-1},\,}$

where ${\displaystyle \scriptstyle A(9,k)|_{k=1}^{9}\,=\,\,}$

${\displaystyle \scriptstyle \{1,502,14608,88234,156190,88234,14608,502,1\}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{8}{\binom {9}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 548\,}$ ${\displaystyle \sum _{i=1}^{9}{\binom {9}{i}}(n-1)^{9-i}\,}$ ${\displaystyle H_{m}^{(-9)}\,}$ ${\displaystyle H_{m}^{(9)}\,}$ ${\displaystyle \zeta (9)\,}$

Base 10: A013667
CFrac: A013685

10 Dekeract numbers

10-orthotope

icosa-9-cell

(..., 20)

{4, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{10}(x)}{(1-x)^{11}}}\,}$

with ${\displaystyle \scriptstyle A_{10}(x)\,=\,\sum _{k=1}^{10}A(10,k)\ x^{k-1},\,}$

where ${\displaystyle \scriptstyle A(10,k)|_{k=1}^{10}\,=\,\,}$

${\displaystyle \scriptstyle \{1,1013,47840,455192,1310354,\,}$

${\displaystyle \scriptstyle 1310354,455192,47840,1013,1\}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{9}{\binom {10}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 1079\,}$ ${\displaystyle \sum _{i=1}^{10}{\binom {10}{i}}(n-1)^{10-i}\,}$ ${\displaystyle H_{m}^{(-10)}\,}$ ${\displaystyle H_{m}^{(10)}\,}$ ${\displaystyle \zeta (10)=\,}$

${\displaystyle {{\pi ^{10}} \over 93555}\,}$

Base 10: A013668
CFrac: A013686

11 Hendekeract numbers

11-orthotope

icosidi-10-cell

(..., 22)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{11}(x)}{(1-x)^{12}}}\,}$

with ${\displaystyle \scriptstyle A_{11}(x)\,=\,\sum _{k=1}^{11}A(11,k)\ x^{k-1},\,}$

where ${\displaystyle \scriptstyle A(11,k)|_{k=1}^{11}\,=\,\,}$

${\displaystyle \scriptstyle \{1,2036,...,2036,1\}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{10}{\binom {11}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 2132\,}$ ${\displaystyle \sum _{i=1}^{11}{\binom {11}{i}}(n-1)^{11-i}\,}$ ${\displaystyle H_{m}^{(-11)}\,}$ ${\displaystyle H_{m}^{(11)}\,}$ ${\displaystyle \zeta (11)\,}$

Base 10: A013669
CFrac: A013687

12 Dodekeract numbers

12-orthotope

icositetra-11-cell

(..., 24)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

${\displaystyle {\frac {x\ A_{12}(x)}{(1-x)^{13}}}\,}$

with ${\displaystyle \scriptstyle A_{12}(x)\,=\,\sum _{k=1}^{12}A(12,k)\ x^{k-1},\,}$

where ${\displaystyle \scriptstyle A(12,k)|_{k=1}^{12}\,=\,\,}$

${\displaystyle \scriptstyle \{1,4083,...,4083,1\}\,}$

${\displaystyle \scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{11}{\binom {12}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,}$

${\displaystyle 4223\,}$ ${\displaystyle \sum _{i=1}^{12}{\binom {12}{i}}(n-1)^{12-i}\,}$ ${\displaystyle H_{m}^{(-12)}\,}$ ${\displaystyle H_{m}^{(12)}\,}$ ${\displaystyle \zeta (12)=\,}$

${\displaystyle {{691\pi ^{12}} \over 638512875}\,}$

Base 10: A013670
CFrac: A013688

Table of sequences

Regular orthotopic numbers sequences
${\displaystyle d\,}$ OEIS

number

${\displaystyle P_{2^{d}}^{(d)}(n),\ n\geq 0\,}$ sequences
0 (NOT A057427) [11] {0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
1 A001477 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ...}
2 A000290 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, ...}
3 A000578 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, ...}
4 A000583 {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, ...}
5 A000584 {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, ...}
6 A001014 {0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, ...}
7 A001015 {0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, ...}
8 A001016 {0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, ...}
9 A001017 {0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, 38443359375, ...}
10 A008454 {0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, 289254654976, ...}
11 A008455 {0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, ...}
12 A008456 {0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, 23298085122481, ...}

Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)

The regular orthotopic numbers read cross-dimensionally give the exponentials sequences

Note the disagreement about 0^0,[22] between the figurate number interpretation (which has to be 0 for ${\displaystyle \scriptstyle n\,=\,0\,}$) and the powers interpretation (which is 1.)

Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)
${\displaystyle b\,}$ OEIS

number

${\displaystyle P_{2^{n}}^{(n)}(b)=b^{n},\ n\geq 0\,}$ sequences
0 [22] A000007 {0[23], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}
1 A000012 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
2 A000079 {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, ...}
3 A000244 {1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, ...}
4 A000302 {1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, ...}
5 A000351 {1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, ...}
6 A000400 {1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...}
7 A000420 {1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, ...}
8 A001018 {1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...}
9 A001019 {1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...}
10 A011557 {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...}
11 A001020 {1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...}
12 A001021 {1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...}

Notes

1. Weisstein, Eric W., Hypercube, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Hypercube.html]
2. Where ${\displaystyle \scriptstyle P_{N_{0}}^{(d)}(n)\,}$ is the ${\displaystyle \scriptstyle d\,}$-dimensional regular convex polytope number with ${\displaystyle \scriptstyle N_{0}\,}$ vertices.
3. Since the power series associated with generating functions are only formal, i.e. used as placeholders for the ${\displaystyle \scriptstyle a_{n}\,}$ as coefficients of ${\displaystyle \scriptstyle x^{n}\,}$, we need not worry about convergence (as long as it converges for some range of ${\displaystyle \scriptstyle x\,}$, whatever that range.)
4. Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.
5. Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld, Generating Functions, Mathematics for Computer Science, MIT, 2005.
6. Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
7. Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/WaringsProblem.html]
8. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HarmonicNumber.html] Cite error: Invalid <ref> tag; name "HarmonicNumber" defined multiple times with different content
9. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RiemannZetaFunction.html]
10. Weisstein, Eric W., Schläfli Symbol, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/SchlaefliSymbol.html]
11. A057427 is the sign function (-1 for n < 0, 0 for n = 0, +1 for n > 0,) while what we get here is the characteristic function of positive integers (0 for n ≤ 0, +1 for n ≥ 1.)
12. Weisstein, Eric W., Nexus Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/NexusNumber.html]
13. Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION, J. Korean Math. Soc. 44 (2007), No. 2, pp. 487-498.
14. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
15. Weisstein, Eric W., Bernoulli Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/BernoulliNumber.html]
16. Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RisingFactorial.html]
17. Weisstein, Eric W., Multichoose, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Multichoose.html]
18. Weisstein, Eric W., Odd Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/OddNumber.html]
19. Weisstein, Eric W., Hex Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HexNumber.html]
20. Weisstein, Eric W., Apéry's Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Ap%C3%A9rysConstant.html]
21. Weisstein, Eric W., Rhombic Dodecahedral Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RhombicDodecahedralNumber.html]
22. Note the disagreement about 0^0 between the figurate number interpretation (which has to be 0 for ${\displaystyle \scriptstyle n\,=\,0\,}$) and the powers interpretation (which is 1.)