Centered simplicial polytopic numbers

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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 $\scriptstyle \mathbb {R} ^{d}\,$, 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 n^th d-dimensional centered simplicial polytopic number is given by the formula:

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

where d is the dimension.

Schläfli-Poincaré (convex) polytope formula

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

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

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

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

with initial conditions

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

Generating function

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

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. 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 $\scriptstyle \{n^{2}|n=0,1,2,\ldots \}\,$ of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every $\scriptstyle k\geq 3$, the set $\scriptstyle \{P(k,n)|n=0,1,2,\ldots \}\,$ 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 $\scriptstyle g(d)\,$ such that every nonnegative integer is a sum of $\scriptstyle g(d)\,$ $\scriptstyle d\,$^th powers, i.e. the set $\scriptstyle \{n^{d}|n=0,1,2,\ldots \}\,$ of $\scriptstyle d\,$^th powers forms a basis of order $\scriptstyle g(d)\,$. The Hilbert-Waring problem is concerned with the study of $\scriptstyle g(d)\,$ for $\scriptstyle d\geq 2\,$. 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

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

Partial sums

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

where $\scriptstyle T_{m}\,$ is the m^th triangular number.

Partial sums of reciprocals

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

Sum of reciprocals

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

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	Formulae $\,_{c}P_{d+1}^{(d)}(n)\,$	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) {}	${\binom {n+2}{2}}-{\binom {n}{2}}\,$ $2n+1\,$ Odd numbers	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}	$\,$	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}	${\binom {n+4}{4}}-{\binom {n}{4}}\,$	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}	$\scriptstyle {\binom {n}{0}}+5{\binom {n}{1}}+10{\binom {n}{2}}+10{\binom {n}{3}}+5{\binom {n}{4}}\,$	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}	${\binom {n+6}{6}}-{\binom {n}{6}}\,$	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}	$\,$	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}	${\binom {n+8}{8}}-{\binom {n}{8}}\,$	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}	$\,$	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}	${\binom {n+10}{10}}-{\binom {n}{10}}\,$	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}	$\,$	1	12	78	364	1365	4368	12376	31824	75582	167960	352716	705431	1352066	A008504
11	Centered dodecadecon 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}	${\binom {n+12}{12}}-{\binom {n}{12}}\,$	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}	$\,$	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	Generating function $G_{\{\,_{c}P_{d+1}^{(d)}\}}(x)=\,$ ${\frac {1-x^{d+1}}{(1-x)^{d+2}}}\,$	Order of basis $g_{\{\,_{c}P_{d+1}^{(d)}\}}\,$	Differences $\,_{c}P_{d+1}^{(d)}(n)-\,$ $\,_{c}P_{d+1}^{(d)}(n-1)=\,$	Partial sums $\sum _{n=0}^{m}{\,_{c}P_{d+1}^{(d)}(n)}=\,$	Partial sums of reciprocals $\sum _{n=0}^{m}{\frac {1}{\,_{c}P_{d+1}^{(d)}(n)}}=\,$	Sum of reciprocals $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{d+1}^{(d)}}(n)}=\,$
1	Centered 2-step gnomonic 1-simplex bi-0-cell (2) {}	${\frac {1-x^{2}}{(1-x)^{3}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
2	Centered triangular 2-simplex tri-1-cell (3, 3) {3}	${\frac {1-x^{3}}{(1-x)^{4}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
3	Centered tetrahedral 3-simplex tetra-2-cell (4, 6, 4) {3, 3}	${\frac {1-x^{4}}{(1-x)^{5}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
4	Centered pentachoron 4-simplex penta-3-cell (5, 10, 10, 5) {3, 3, 3}	${\frac {1-x^{5}}{(1-x)^{6}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
5	Centered hexateron 5-simplex hexa-4-cell (6, 15, 20, 15, 6) {3, 3, 3, 3}	${\frac {1-x^{6}}{(1-x)^{7}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
6	Centered heptapeton 6-simplex hepta-5-cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3}	${\frac {1-x^{7}}{(1-x)^{8}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
7	Centered octahexon 7-simplex octa-6-cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3}	${\frac {1-x^{8}}{(1-x)^{9}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
8	Centered enneahepton 8-simplex nona-7-cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3}	${\frac {1-x^{9}}{(1-x)^{10}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
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}	${\frac {1-x^{10}}{(1-x)^{11}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
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}	${\frac {1-x^{11}}{(1-x)^{12}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
11	Centered dodecadecon 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}	${\frac {1-x^{12}}{(1-x)^{13}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
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}	${\frac {1-x^{13}}{(1-x)^{14}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$

Table of sequences

Centered simplicial polytopic numbers sequences

d	$\,_{c}P_{d+1}^{(d)}(n),\ n\geq 0\,$ 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, ...}

See also

Simplicial polytopic numbers

External links

• S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
• Herbert S. Wilf, generatingfunctionology, 2^nd ed., 1994.