Simplicial polytopic numbers

The simplicial polytopic numbers are a family of sequences of 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 ${\mathbb{ℝ}}^d$, 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 simplicial d-polytopic numbers are given by the formulae :

$P_{d+1}^{(d)}(n)=(\genfrac{}{}{0ex}{}{d+(n-1)}{d})=\frac{d^{(n)}}{d!}$, or

$P_{d+1}^{(d)}(n)=(\genfrac{}{}{0ex}{}{n+(d-1)}{d})=\frac{n^{(d)}}{d!}=\left((\genfrac{}{}{0ex}{}{n}{d})\right),$

where d ≥ 0 is the dimension and n-1 ≥ 0 is the number of nondegenerate layered simplices (n-1 = 0 giving a single dot, a degenerate simplex) of the d-dimensional regular convex simplicial polytope number (d-simplex number.)

Recurrence relation

$P_{d+1}^{(d)}(n)=P_{d+1}^{(d)}(n-1)+P_d^{(d-1)}(n)$

Generating function

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

Simplicial polytopic numbers and Pascal's (rectangular) triangle columns

Pascal's triangle

n = 0	1
1	1	1
2	1	2	1
3	1	3	3	1
4	1	4	6	4	1
5	1	5	10	10	5	1
6	1	6	15	20	15	6	1
7	1	7	21	35	35	21	7	1
8	1	8	28	56	70	56	28	8	1
9	1	9	36	84	126	126	84	36	9	1
10	1	10	45	120	210	252	210	120	45	10	1
11	1	11	55	165	330	462	462	330	165	55	11	1
12	1	12	66	220	495	792	924	792	495	220	66	12	1
	d = 0	1	2	3	4	5	6	7	8	9	10	11	12

The d^th column of Pascal's triangle gives the d-dimensional simplicial polytopic numbers, forming simplex polytopes, where (-1)-cells correspond to the empty set, 0-cells are vertices, 1-cells are edges, 2-cells are faces, and so on...
 

d = 0	0-simplicial numbers	Point numbers	Form point	(1 (-1)-cells "faces")	(0-simplex)
d = 1	1-simplicial numbers	Linear numbers	Form segments	(2 0-cells "faces")	(1-simplex)
d = 2	2-simplicial numbers	Triangular numbers	Form triangles	(3 1-cells "faces")	(2-simplex)
d = 3	3-simplicial numbers	Tetrahedral numbers	Form tetrahedrons	(4 2-cells "faces")	(3-simplex)
d = 4	4-simplicial numbers	Pentachoron numbers	Form pentachorons	(5 3-cells "faces")	(4-simplex)
d = 5	5-simplicial numbers	Hexateron numbers	Form hexaterons	(6 4-cells "faces")	(5-simplex)
d = 6	6-simplicial numbers	Heptapeton numbers	Form heptapetons	(7 5-cells "faces")	(6-simplex)
d = 7	7-simplicial numbers	Octahexon numbers	Form octahexons	(8 6-cells "faces")	(7-simplex)
d = 8	8-simplicial numbers	Nonahepton numbers	Form nonaheptons	(9 7-cells "faces")	(8-simplex)

Order of basis

See also Polygonal numbers#Order of basis

For a subset of nonnegative integers $S$, its order as a basis is defined as the minimal number $g$ such that all integers are representable as at least $g$ members of $S$.

Whereas the $k$-gonal numbers form a basis of order $k$ (the polygonal number theorem), and the $d$-dimensional hypercubic numbers form a basis of finite order dependent on $d$ (the Hilbert-Waring theorem), relatively little is known about the simplicial numbers. Pollock's Conjecture states that the tetrahedral numbers are a basis of order 5. A000797 gives the integers known not to be writeable as a sum of four, and A282172(n) is the number of ways to write $n$ as an ordered sum of five.

5 is the first number not a sum of 2 triangular numbers, 17 the first not a sum of 4 tetrahedrals, 64 the first not a sum of 7 pentachorals, 220 the first not a sum of 9 hexaterals, 839 the first not a sum of 12 heptapetals. These first six terms suggest the smallest number not writeable as $\mathrm{o}\mathrm{r}\mathrm{d}(P_{d+1}^{(d)})-1$ $d$-simplexes is $A366978(d)=\sum _{j=1}^d{P}_{d+2}^{(d+1)}\left(\lfloor \frac{d}{j}\rfloor \right)=\sum _{j=1}^d(\genfrac{}{}{0ex}{}{d+\lfloor \frac{d}{j}\rfloor }{d+1})$, but this pattern breaks down with 3137 being the first not writeable as 14 octahexal numbers.

Differences

$P_{d+1}^{(d)}(n)-P_{d+1}^{(d)}(n-1)=P_d^{(d-1)}(n)$

Partial sums

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

Partial sums of reciprocals

$\sum _{n=1}^m\frac{1}{P_{d+1}^{(d)}(n)}=\frac{d-\frac{d+m}{P_{d+1}^{(d)}(m+1)}}{d-1}$

We can prove this by writing $P_{d+1}^{(d)}(n)=(\genfrac{}{}{0ex}{}{n+d-1}{d})=\frac{n^{\bar{d}}}{d!}$ (where $n^{\bar{d}}$ is the rising factorial)

$=d!\sum _{n=1}^m\frac{1}{n^{\bar{d}}}$

and using the fact that $\frac{1}{n^{\bar{d}}}=\frac{(n-1)!}{(n-1+d)!}=(n-1{)}^{\underset{\_}{-d}}$

$=d!\sum _{n=1}^m(n-1{)}^{\underset{\_}{-d}}=d!\sum _{n=0}^{m-1}{n}^{\underset{\_}{-d}}$

and since (in analogy with monomial integration) we have that $\sum _{n=a}^{b-1}{n}^{\underset{\_}{d}}=\{\begin{array}{ll}\frac{b^{\underset{\_}{d+1}}-a^{\underset{\_}{d+1}}}{d+1}& d\ne -1\\ H_b-H_a& d=-1\end{array}$ (see Concrete Mathematics, page 53, eq. (2.53)), this is

$\begin{array}{rl}& =d!\frac{m^{\underset{\_}{1-d}}-0^{\underset{\_}{1-d}}}{1-d}\\ & =d(d-2)!(0^{\underset{\_}{1-d}}-m^{\underset{\_}{1-d}})\\ & =d(d-2)!(\frac{1}{1^{\overline{d-1}}}-\frac{1}{(m+1{)}^{\overline{d-1}}})\\ & =d(d-2)!(\frac{1}{(d-1)!}-\frac{m!}{(m+d-1)!})\\ & =\frac{d-\frac{d}{(\genfrac{}{}{0ex}{}{m+d-1}{m})}}{d-1}=\frac{d-\frac{d+m}{(\genfrac{}{}{0ex}{}{m+d}{m})}}{d-1}\end{array}$

which, in terms of simplicial numbers, are $\frac{d-\frac{d}{P_d^{(d-1)}(m+1)}}{d-1}=\frac{d-\frac{d+m}{P_{d+1}^{(d)}(m+1)}}{d-1}$

Sum of reciprocals

The sum of reciprocals $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{P_{d+1}^{(d)}(n)}=\frac{d}{d-1}=\frac{1}{1-\frac{1}{d}}$ can be interpreted as $\frac{1}{p}$, where $p=1-\frac{1}{d}$ is the probability that $d$ does not divide a random integer $x$ or the probability that two random integers $x$ and $y$ have different residues modulo $d$.

The reciprocal of the sum of reciprocals thus gives:

$\frac{1}{\sum _{n=1}^{\mathrm{\infty }}\frac{1}{P_{d+1}^{(d)}(n)}}=\frac{d-1}{d}=1-\frac{1}{d}=P(x\equiv ̸0(\mathrm{mod}d))=P(x\equiv ̸y(\mathrm{mod}d))$

Graphically, the second interpretation is the probability that a 2-D integer lattice point $(x,y)$ does not lay on any straight line of the family $y=x+kd$, for integer values of $k$. This family of functions consists of the bisection of the first and third quadrants and all its parallels at distances $\frac{kd}{\sqrt{2}}$. When $d=1$ this includes all lattice points, hence $p=0$.

Number of j-dimensional "vertices"

$N_j=(\genfrac{}{}{0ex}{}{d+1}{j+1}),(0\le j\le d)$

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 simplicial polytopic numbers are listed by increasing number N₀ of vertices.

Simplicial numbers formulae and values

d	Name d-simplex d+1 (d-1)-cell (N0, N1, N2, ...) Schläfli symbol	Formulae $P_{d+1}^{(d)}(n)$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
0	Point 0-simplex hena-(-1)-cell () {}	$(\genfrac{}{}{0ex}{}{n-1}{0})$	0	1	1	1	1	1	1	1	1	1	1	1	1	(NOT A057427)
1	Triangular gnomon 1-simplex di-0-cell (2) {}	$(\genfrac{}{}{0ex}{}{n}{1})$ $\frac{n^{(1)}}{1!}$	0	1	2	3	4	5	6	7	8	9	10	11	12	A000027(n)
2	Triangular 2-simplex tri-1-cell (3, 3) {3}	$(\genfrac{}{}{0ex}{}{n+1}{2})$ $\frac{n^{(2)}}{2!}$	0	1	3	6	10	15	21	28	36	45	55	66	78	A000217(n)
3	Tetrahedral 3-simplex tetra-2-cell (4, 6, 4) {3, 3}	$(\genfrac{}{}{0ex}{}{n+2}{3})$ $\frac{n^{(3)}}{3!}$	0	1	4	10	20	35	56	84	120	165	220	286	364	A000292(n)
4	Pentachoron 4-simplex penta-3-cell (5, 10, 10, 5) {3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+3}{4})$ $\frac{n^{(4)}}{4!}$	0	1	5	15	35	70	126	210	330	495	715	1001	1365	A000332(n+3)
5	Hexateron 5-simplex hexa-4-cell (6, 15, 20, 15, 6) {3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+4}{5})$ $\frac{n^{(5)}}{5!}$	0	1	6	21	56	126	252	462	792	1287	2002	3003	4368	A000389(n+4)
6	Heptapeton 6-simplex hepta-5-cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+5}{6})$ $\frac{n^{(6)}}{6!}$	0	1	7	28	84	210	462	924	1716	3003	5005	8008	12376	A000579(n+5)
7	Octahexon 7-simplex octa-6-cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+6}{7})$ $\frac{n^{(7)}}{7!}$	0	1	8	36	120	330	792	1716	3432	6435	11440	19448	31824	A000580(n+6)
8	Enneahepton 8-simplex nona-7-cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+7}{8})$ $\frac{n^{(8)}}{8!}$	0	1	9	45	165	495	1287	3003	6435	12870	24310	43758	75582	A000581(n+7)
9	Decaocton 9-simplex deca-8-cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+8}{9})$ $\frac{n^{(9)}}{9!}$	0	1	10	55	220	715	2002	5005	11440	24310	48620	92378	167960	A000582(n+8)
10	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}	$(\genfrac{}{}{0ex}{}{n+9}{10})$ $\frac{n^{(10)}}{10!}$	0	1	11	66	286	1001	3003	8008	19448	43758	92378	184756	352716	A001287(n+9)
11	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}	$(\genfrac{}{}{0ex}{}{n+10}{11})$ $\frac{n^{(11)}}{11!}$	0	1	12	78	364	1365	4368	12376	31824	75582	167960	352716	705432	A001288(n+10)
12	Tridecahendecon 12-simplex trideca-11-cell (13, ... Pascal's triangle 13th row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}	$(\genfrac{}{}{0ex}{}{n+11}{12})$ $\frac{n^{(12)}}{12!}$	0	1	13	91	455	1820	6188	18564	50388	125970	293930	646646	1352078	A010965(n+11)

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 simplicial polytopic numbers are listed by increasing number N₀ of vertices.

Simplicial numbers related formulae and values

d	Name d-simplex d+1 (d-1)-cell (N0, N1, N2, ...) Schläfli symbol	Generating function $G_{\{P_{d+1}^{(d)}\}}(x)=$ $\frac{x}{(1-x{)}^{d+1}}$	Order of basis $g_{\{P_{d+1}^{(d)}\}}=$ $N_0+?$	Differences $P_{d+1}^{(d)}(n)-$ $P_{d+1}^{(d)}(n-1)=$ $P_d^{(d-1)}(n)$	Partial sums $\sum _{n=1}^m{P}_{d+1}^{(d)}(n)=$ $P_{d+2}^{(d+1)}(m)$	Partial sums of reciprocals $\sum _{n=1}^m\frac{1}{P_{d+1}^{(d)}(n)}=$ $\frac{d-\frac{m+d}{(\genfrac{}{}{0ex}{}{m+d}{d})}}{d-1}$	Sum of reciprocals $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{P_{d+1}^{(d)}(n)}=$ $\frac{d}{d-1}$
1	Triangular gnomon 1-simplex bi-0-cell (2) {}	$\frac{x}{(1-x{)}^2}$	$1$ $(2d-1)$	$$	$$	$H_m$ [1] A001008(m)/A002805(m) (reduced) A000254(m)/A000142(m)	$\underset{m\to \mathrm{\infty }}{\lim }{H}_m\sim \log (m)\to \mathrm{\infty }$
2	Triangular 2-simplex tri-1-cell (3, 3) {3}	$\frac{x}{(1-x{)}^3}$	$3$ $(2d-1)$	$$	$$	$\frac{2(\genfrac{}{}{0ex}{}{m+2}{2})-m-2}{1(\genfrac{}{}{0ex}{}{m+2}{2})}$ [2] = $\frac{2m}{(m+1)}$	$2$
3	Tetrahedral 3-simplex tetra-2-cell (4, 6, 4) {3, 3}	$\frac{x}{(1-x{)}^4}$	$5?$ $(2d-1)?$	$$	$$	$\frac{3(\genfrac{}{}{0ex}{}{m+3}{3})-m-3}{2(\genfrac{}{}{0ex}{}{m+3}{3})}$ [3] = $\frac{3(m^2+3m)}{2(m^2+3m+2)}$	$\frac{3}{2}$ [4]
4	Pentachoron 4-simplex penta-3-cell (5, 10, 10, 5) {3, 3, 3}	$\frac{x}{(1-x{)}^5}$	$8?$ $(2d)?$	$$	$$	$\frac{4(\genfrac{}{}{0ex}{}{m+4}{4})-m-4}{3(\genfrac{}{}{0ex}{}{m+4}{4})}$ [5] = $\frac{4(m^3+6m^2+11m)}{3(m^3+6m^2+11m+6)}$ A118411/A118412	$\frac{4}{3}$
5	Hexateron 5-simplex hexa-4-cell (6, 15, 20, 15, 6) {3, 3, 3, 3}	$\frac{x}{(1-x{)}^6}$	$10?$ $(2d)?$	$$	$$	$\frac{5(\genfrac{}{}{0ex}{}{m+5}{5})-m-5}{4(\genfrac{}{}{0ex}{}{m+5}{5})}$ [6]	$\frac{5}{4}$
6	Heptapeton 6-simplex hepta-5-cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3}	$\frac{x}{(1-x{)}^7}$	$13?$ $(2d+1)?$	$$	$$	$\frac{6(\genfrac{}{}{0ex}{}{m+6}{6})-m-6}{5(\genfrac{}{}{0ex}{}{m+6}{6})}$ [7]	$\frac{6}{5}$
7	Octahexon 7-simplex octa-6-cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3}	$\frac{x}{(1-x{)}^8}$	$15?$ $(2d+1)?$	$$	$$	$\frac{7(\genfrac{}{}{0ex}{}{m+7}{7})-m-7}{6(\genfrac{}{}{0ex}{}{m+7}{7})}$ [8]	$\frac{7}{6}$
8	Enneahepton 8-simplex nona-7-cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3}	$\frac{x}{(1-x{)}^9}$	$(2d+1)?$	$$	$$	$\frac{8(\genfrac{}{}{0ex}{}{m+8}{8})-m-8}{7(\genfrac{}{}{0ex}{}{m+8}{8})}$ [9]	$\frac{8}{7}$
9	Decaocton 9-simplex deca-8-cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3}	$\frac{x}{(1-x{)}^{10}}$	$(2d+1)?$	$$	$$	$\frac{9(\genfrac{}{}{0ex}{}{m+9}{9})-m-9}{8(\genfrac{}{}{0ex}{}{m+9}{9})}$ [10]	$\frac{9}{8}$
10	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}	$\frac{x}{(1-x{)}^{11}}$	$(2d+1)?$	$$	$$	$\frac{10(\genfrac{}{}{0ex}{}{m+10}{10})-m-10}{9(\genfrac{}{}{0ex}{}{m+10}{10})}$ [11]	$\frac{10}{9}$
11	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}	$\frac{x}{(1-x{)}^{12}}$	$(2d+1)?$	$$	$$	$\frac{11(\genfrac{}{}{0ex}{}{m+11}{11})-m-11}{10(\genfrac{}{}{0ex}{}{m+11}{11})}$ [12]	$\frac{11}{10}$
12	Tridecahendecon 12-simplex trideca-11-cell (13, ... Pascal's triangle 13th row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}	$\frac{x}{(1-x{)}^{13}}$	$(2d+1)?$	$$	$$	$\frac{12(\genfrac{}{}{0ex}{}{m+12}{12})-m-12}{11(\genfrac{}{}{0ex}{}{m+12}{12})}$ [13]	$\frac{12}{11}$

Table of sequences

Simplicial polytopic numbers sequences

d	$P_{d+1}^{(d)}(n),n\ge 0$ sequences
1	{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	{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, ...}
3	{0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, ...}
4	{0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, ...}
5	{0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, ...}
6	{0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, ...}
7	{0, 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, ...}
8	{0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, ...}
9	{0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, ...}
10	{0, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, ... }
11	{0, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, ...}
12	{0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, ...}

See also

Centered 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.