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# Centered simplicial polytopic numbers

(Redirected from Centered simplex numbers)

The centered simplicial polytopic numbers are a family of sequences of centered figurate numbers corresponding to the d-dimensional simplex for each dimension d, where d is a nonnegative integer.

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

## Minimal nondegenerate polytopes in a d-dimensional Euclidean space, d ≥ 0

In a d-dimensional Euclidean space in ${\displaystyle \scriptstyle \mathbb {R} ^{d}\,}$, d ≥ 0, the minimal number of vertices d + 1 gives the simplest d-polytope (the d-simplex,) i.e.:

• d = 0: the 0-simplex (having 1 vertex) is the point (the 1 (-1)-cell, with 1 null polytope as facet)
• d = 1: the 1-simplex (having 2 vertices) is the triangular gnomon (the 2 0-cell, with 2 points as facets)
• d = 2: the 2-simplex (having 3 vertices) is the trigon (triangle) (the 3 1-cell, with 3 segments as facets)
• d = 3: the 3-simplex (having 4 vertices) is the tetrahedron (the 4 2-cell, with 4 faces as facets)
• d = 4: the 4-simplex (having 5 vertices) is the pentachoron (the 5 3-cell, with 5 rooms as facets)
• d = 5: the 5-simplex (having 6 vertices) is the hexateron (the 6 4-cell, with 6 4-cells as facets)
• d = 6: the 6-simplex (having 7 vertices) is the heptapeton (the 7 5-cell, with 7 5-cells as facets)
• d = 7: the 7-simplex (having 8 vertices) is the octahexon (the 8 6-cell, with 8 6-cells as facets)
• d = 8: the 8-simplex (having 9 vertices) is the enneahepton (the 9 7-cell, with 9 7-cells as facets)
• ...
• d = d: the d-simplex (having d+1 vertices) is the d+1 (d-1)-cell, with d+1 (d-1)-cells as facets

## Formulae

The nth d-dimensional centered simplicial polytopic number is given by the formula:

${\displaystyle \,_{c}P_{d+1}^{(d)}(n)=?,\,}$ [1]

where d is the dimension.

## Schläfli-Poincaré (convex) polytope formula

Generalization for polytopes of Descartes-Euler (convex) polyhedral formula:[2]

${\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=1-(-1)^{d},\,}$

where N0 is the number of 0-dimensional elements, N1 is the number of 1-dimensional elements, N2 is the number of 2-dimensional elements...

## Recurrence equation

${\displaystyle \,_{c}P_{d+1}^{(d)}(n)=?,\,}$

with initial conditions

${\displaystyle \,_{c}P_{d+1}^{(d)}(0)=1,\,_{c}P_{d+1}^{(d)}(1)=?,\,_{c}P_{d+1}^{(d)}(2)=?\,}$

## Generating function

${\displaystyle G_{\{\,_{c}P_{d+1}^{(d)}\}}(x)={\frac {1-x^{d+1}}{(1-x)^{d+2}}}\,}$

## 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 k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[3] 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 k k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)

A nonempty subset A of nonnegative integers is called a basis of order g if g is the minimum number with the property that every nonnegative integer can be written as a sum of g elements in 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 k-gon numbers forms a basis of order k, i.e. every nonnegative integer can be written as a sum of k k-gon 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.

## Differences

${\displaystyle \,_{c}P_{d+1}^{(d)}(n)-\,_{c}P_{d+1}^{(d)}(n-1)=?\,}$

## Partial sums

${\displaystyle \sum _{n=0}^{m}\,_{c}P_{d+1}^{(d)}(n)=?,\,}$

where ${\displaystyle \scriptstyle T_{m}\,}$ is the mth triangular number.

## Partial sums of reciprocals

${\displaystyle \sum _{n=0}^{m}{\frac {1}{\,_{c}P_{d+1}^{(d)}(n)}}=?\,}$

## Sum of reciprocals

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\,_{c}P_{d+1}^{(d)}(n)}}=?\,}$

## Table of formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The centered simplicial polytopic numbers are listed by increasing number N0 of vertices.

Centered simplicial numbers formulae and values
d Name

d-simplex

d+1 (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[4]

Formulae

${\displaystyle \,_{c}P_{d+1}^{(d)}(n)\,}$

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

1 Centered 2-step gnomonic

1-simplex

bi-0-cell

(2)

{}

${\displaystyle {\binom {n+2}{2}}-{\binom {n}{2}}\,}$

${\displaystyle 2n+1\,}$

1 3 5 7 9 11 13 15 17 19 21 23 25 A005408
2 Centered triangular

2-simplex

tri-1-cell

(3, 3)

{3}

${\displaystyle \,}$ 1 4 10 19 31 46 64 85 109 136 166 199 235 A005448
3 Centered tetrahedral

3-simplex

tetra-2-cell

(4, 6, 4)

{3, 3}

${\displaystyle {\binom {n+4}{4}}-{\binom {n}{4}}\,}$ 1 5 15 35 69 121 195 295 425 589 791 1035 1325 A005894
4 Centered pentachoron

4-simplex

penta-3-cell

(5, 10, 10, 5)

{3, 3, 3}

${\displaystyle \scriptstyle {\binom {n}{0}}+5{\binom {n}{1}}+10{\binom {n}{2}}+10{\binom {n}{3}}+5{\binom {n}{4}}\,}$ 1 6 21 56 126 251 456 771 1231 1876 2751 3906 5396 A008498
5 Centered hexateron

5-simplex

hexa-4-cell

(6, 15, 20, 15, 6)

{3, 3, 3, 3}

${\displaystyle {\binom {n+6}{6}}-{\binom {n}{6}}\,}$ 1 7 28 84 210 462 923 1709 2975 4921 7798 11914 17640 A008499
6 Centered heptapeton

6-simplex

hepta-5-cell

(7, 21, 35, 35, 21, 7)

{3, 3, 3, 3, 3}

${\displaystyle \,}$ 1 8 36 120 330 792 1716 3431 6427 11404 19328 31494 49596 A008500
7 Centered octahexon

7-simplex

octa-6-cell

(8, 28, 56, 70, 56, 28, 8)

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

${\displaystyle {\binom {n+8}{8}}-{\binom {n}{8}}\,}$ 1 9 45 165 495 1287 3003 6435 12869 24301 43713 75417 125475 A008501
8 Centered enneahepton

8-simplex

nona-7-cell

(9, 36, 84, 126, 126, 84, 36, 9)

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

${\displaystyle \,}$ 1 10 55 220 715 2002 5005 11440 24310 48619 92368 167905 293710 A008502
9 Centered decaocton

9-simplex

deca-8-cell

(10, 45, 120, 210, 252, 210, 120, 45, 10)

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

${\displaystyle {\binom {n+10}{10}}-{\binom {n}{10}}\,}$ 1 11 66 286 1001 3003 8008 19448 43758 92378 184755 352705 646580 A008503
10 Centered hendecaenneon

10-simplex

hendeca-9-cell

(11, 55, 165, 330, 462, 462, 330, 165, 55, 11)

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

${\displaystyle \,}$ 1 12 78 364 1365 4368 12376 31824 75582 167960 352716 705431 1352066 A008504

11-simplex

dodeca-10-cell

(12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12)

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

${\displaystyle {\binom {n+12}{12}}-{\binom {n}{12}}\,}$ 1 13 91 455 1820 6188 18564 50388 125970 293930 646646 1352078 2704155 A008505
12 Centered tridecahendecon

12-simplex

trideca-11-cell

(13, ... Pascal's triangle 13th row..., 13)

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

${\displaystyle \,}$ 1 14 105 560 2380 8568 27132 77520 203490 497420 1144066 2496144 5200300 A008506

## Table of related formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The centered simplicial polytopic numbers are listed by increasing number N0 of vertices.

Centered simplicial numbers related formulae and values
d Name

d-simplex

d+1 (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[4]

Generating

function

${\displaystyle G_{\{\,_{c}P_{d+1}^{(d)}\}}(x)=\,}$

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

Order

of basis

${\displaystyle g_{\{\,_{c}P_{d+1}^{(d)}\}}\,}$ [3][5][6]

Differences

${\displaystyle \,_{c}P_{d+1}^{(d)}(n)-\,}$

${\displaystyle \,_{c}P_{d+1}^{(d)}(n-1)=\,}$

Partial sums

${\displaystyle \sum _{n=0}^{m}{\,_{c}P_{d+1}^{(d)}(n)}=\,}$

Partial sums of reciprocals

${\displaystyle \sum _{n=0}^{m}{\frac {1}{\,_{c}P_{d+1}^{(d)}(n)}}=\,}$

Sum of reciprocals[7]

${\displaystyle \sum _{n=0}^{\infty }{1 \over {\,_{c}P_{d+1}^{(d)}}(n)}=\,}$

1 Centered 2-step gnomonic

1-simplex

bi-0-cell

(2)

{}

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

2-simplex

tri-1-cell

(3, 3)

{3}

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

3-simplex

tetra-2-cell

(4, 6, 4)

{3, 3}

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

4-simplex

penta-3-cell

(5, 10, 10, 5)

{3, 3, 3}

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

5-simplex

hexa-4-cell

(6, 15, 20, 15, 6)

{3, 3, 3, 3}

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

6-simplex

hepta-5-cell

(7, 21, 35, 35, 21, 7)

{3, 3, 3, 3, 3}

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

7-simplex

octa-6-cell

(8, 28, 56, 70, 56, 28, 8)

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

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

8-simplex

nona-7-cell

(9, 36, 84, 126, 126, 84, 36, 9)

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

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

9-simplex

deca-8-cell

(10, 45, 120, 210, 252, 210, 120, 45, 10)

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

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

10-simplex

hendeca-9-cell

(11, 55, 165, 330, 462, 462, 330, 165, 55, 11)

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

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

11-simplex

dodeca-10-cell

(12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12)

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

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

12-simplex

trideca-11-cell

(13, ... Pascal's triangle 13th row..., 13)

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

${\displaystyle {\frac {1-x^{13}}{(1-x)^{14}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$

## Table of sequences

Centered simplicial polytopic numbers sequences
d ${\displaystyle \,_{c}P_{d+1}^{(d)}(n),\ n\geq 0\,}$ sequences
2 {1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, ...}
3 {1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, ...}
4 {1, 6, 21, 56, 126, 251, 456, 771, 1231, 1876, 2751, 3906, 5396, 7281, 9626, 12501, 15981, 20146, 25081, 30876, 37626, 45431, 54396, 64631, 76251, 89376, 104131, ...}
5 {1, 7, 28, 84, 210, 462, 923, 1709, 2975, 4921, 7798, 11914, 17640, 25416, 35757, 49259, 66605, 88571, 116032, 149968, 191470, 241746, 302127, 374073, 459179, 559181, ...}
6 {1, 8, 36, 120, 330, 792, 1716, 3431, 6427, 11404, 19328, 31494, 49596, 75804, 112848, 164109, 233717, 326656, 448876, 607412, 810510, 1067760, 1390236, 1790643, ...}
7 {1, 9, 45, 165, 495, 1287, 3003, 6435, 12869, 24301, 43713, 75417, 125475, 202203, 316767, 483879, 722601, 1057265, 1518517, 2144493, 2982135, 4088655, 5533155, ...}
8 {1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48619, 92368, 167905, 293710, 496705, 815188, 1302499, 2031535, 3100240, 4638205, 6814522, 9847045, 14013220, ...}
9 {1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184755, 352705, 646580, 1143780, 1960255, 3265757, 5303727, 8416837, 13079352, 19937632, 29860259, 43999449, ...}
10 {1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705431, 1352066, 2496066, 4457036, 7724795, 13033527, 21461804, 34565466, 54551718, 84504355, ...}
11 {1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704155, 5200287, 9657609, 17383405, 30419935, 51889747, 86474661, 141070137, ...}
12 {1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400599, 20058286, 37442055, 67863355, 119757470, 206244507, 347346468, ...}

1. Where ${\displaystyle \scriptstyle \,_{c}P_{N_{0}}^{(d)}(n)\,}$ is the d-dimensional centered regular convex polytope number with N0 vertices.