(Centered squares) hyperpyramidal numbers

The (centered squares) hyperpyramidal numbers are starting with a centered square for dimension d = 2, then do a pyramidal stacking for each extra dimension.

Although the Simplicial polytopic numbers are triangular hyperpyramidal numbers, the centered simplicial polytopic numbers are NOT (centered triangles) hyperpyramidal numbers.

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

Formulae

$\,_{'c'}Y_{d+2}^{(d)}(n)=?\,$

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.

For d-dimensional convex polytopes:

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

where N₀ is the number of 0-dimensional elements (vertices V), N₁ is the number of 1-dimensional elements (edges V), N₂ is the number of 2-dimensional elements (faces F) and N₃ is the number of 3-dimensional elements (cells C), and so on... of the convex polytope.

Recurrence equation

$\,_{'c'}Y_{d+2}^{(d)}(n)=?,\,$

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

Generating function

$G_{\{\,_{'c'}Y_{d+2}^{(d)}\}}(x)=?\,$

Order of basis

$g_{\{\,_{'c'}Y_{d+2}^{(d)}\}}=?\,$

Differences

$\,_{'c'}Y_{d+2}^{(d)}(n)-\,_{'c'}Y_{d+2}^{(d)}(n-1)=?\,$

Partial sums

$\sum _{n=1}^{m}\,_{'c'}Y_{d+2}^{(d)}(n)=\,$

Partial sums of reciprocals

$\sum _{n=1}^{m}{\frac {1}{\,_{'c'}Y_{d+2}^{(d)}(n)}}=\,$

Sum of reciprocals

$\sum _{n=1}^{\infty }{\frac {1}{\,_{'c'}Y_{d+2}^{(d)}(n)}}=\,$

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 squares) hyperpyramidal numbers are listed by increasing number N₀ of vertices.

(Centered squares) hyperpyramidal numbers formulae and values

d	Name (N0, N1, N2, ...) Schläfli symbol	Formulae $\,_{'c'}Y_{d+2}^{(d)}(n)=\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
3	Centered square pyramidal (, , ) {, }	${\frac {n(2n^{2}+1)}{3}}\,$ $P_{2\cdot 3}^{(3)}(n)\,$ Octahedral numbers	0	1	6	19	44	85	146	231	344	489	670	891	1156	A005900(n)
4	(, , , ) {, , }	$\,$	1
5	(, , , , ) {, , , }	$\,$	1
6	(, , , , , ) {, , , , }	$\,$	1
7	(, , , , , , ) {, , , , , }	$\,$	1
8	(, , , , , , , ) {, , , , , , }	$\,$	1
9	(, , , , , , , , ) {, , , , , , , }	$\,$	1
10	(, , , , , , , , , ) {, , , , , , , , }	$\,$	1
11	(, , , , , , , , , , ) {, , , , , , , , , }	$\,$	1
12	(, , , , , , , , , , , ) {, , , , , , , , , , }	$\,$	1

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 elements are the actual facets. The (centered squares) hyperpyramidal numbers are listed by increasing number N₀ of vertices.

(Centered squares) hyperpyramidal numbers related formulae and values

d	Name (N0, N1, N2, ...) Schläfli symbol	Generating function $G_{\{\,_{'c'}Y_{d+2}^{(d)}\}}(x)\,$	Order of basis	Differences $\,_{'c'}Y_{d+2}^{(d)}(n)-\,$ $\,_{'c'}Y_{d+2}^{(d)}(n-1)=\,$ $\,_{'c'}Y_{d+1}^{(d-1)}(n)\,$	Partial sums $\sum _{n=1}^{m}{\,_{'c'}Y_{d+2}^{(d)}(n)}=$ $\,_{'c'}Y_{d+3}^{(d+1)}(m)\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {\,_{'c'}Y_{d+2}^{(d)}(n)}}=$	Sum of reciprocals $\sum _{n=1}^{\infty }{1 \over {\,_{'c'}Y_{d+2}^{(d)}(n)}}=$
3	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
4	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
5	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
6	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
7	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
8	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
9	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
10	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
11	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$
12	$\,$	$\,$	$\,$	$\,$	$\,$	$\,$

Table of sequences

(Centered squares) hyperpyramidal numbers sequences

d	$\,_{'c'}Y_{d+2}^{(d)}(n),\ n\geq 0\,$ sequences
3	{0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, ...}
4	{0, 1, ...}
5	{0, 1, ...}
6	{0, 1, ...}
7	{0, 1, ...}
8	{0, 1, ...}
9	{0, 1, ...}
10	{0, 1, ...}
11	{0, 1, ...}
12	{0, 1, ...}

See also

Square hyperpyramidal 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.