Regular polychoron numbers

There are six regular convex polychora (4-dimensional hyper-solids,) which, except for the 24-cell, are the analogues of the Platonic solids. Listed by increasing number of vertices, the six regular convex polychora are:

• the 5 vertices (self dual) 5-cell (pentachoron) or 4-simplex (a hyper-tetrahedron,)
• the 8 vertices 16-cell or 4-cross polytope or 4-orthoplex (a hyper-octahedron,)
• the 16 vertices 8-cell or 4-cube or 4-orthotope or tesseract (a hyper-cube,)
• the 24 vertices (self dual) 24-cell, which has no perfect analogy in higher or lower dimensional spaces,
• the 120 vertices 600-cell (a hyper-icosahedron,)
• and the 600 vertices 120-cell (a hyper-dodecahedron.)

The 5-cell and 24-cell are self-dual, the 16-cell is the dual of the 8-cell, and the 600- and 120-cells are dual to each other.

The number of cells, faces, edges and vertices, for each of the six regular convex polychora are give the sequences:

• A063924 Number of cells (3-dimensional elements) in the regular 4-dimensional polytopes.
• A063925 Number of faces (2-dimensional elements) in the regular 4-dimensional polytopes.
• A063926 Number of edges (1-dimensional elements) in the six regular 4-dimensional polytopes.
• A063927 Number of vertices (0-dimensional elements) in the regular 4-dimensional polytopes.

The polychoron numbers are the numbers of dots in a layered geometric arrangement into one of the 6 regular convex polychoron hyper-solids. The polychoron numbers start with one initial dot (n = 1,) then with one dot at each vertex of a given polychoron hyper-solid (n = 2,) with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots are counted only once.

The 6 types of regular convex polychoron numbers are:

• A000332(n+3) Pentachoron numbers, binomial(n+3,4).
• A000583 The tesseract numbers, fourth powers: n^4.
• A092182 Figurate numbers based on the 600-cell (4-D polytope with Schlaefli symbol {3,3,5}).
• A092183 Figurate numbers based on the 120-cell (4-D polytope with Schlaefli symbol {5,3,3}).
• A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).

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

Formulae

The n^th 4-dimensional N₃-cell regular polytope (having N₀ vertices) number is given by the formula:

$P_{N_{0}}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[???]{\binom {n+1}{4}}+[???]{\binom {n}{4}},\,$

For the 5-cell numbers:

$P_{5}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[0]{\binom {n}{4}},\,$

For the 16-cell and 8-cell numbers:

$P_{8}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[1]{\binom {n}{4}},\,$

$P_{16}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[1]{\binom {n}{4}},\,$

For the 24-cell numbers:

$P_{24}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[43]{\binom {n+1}{4}}+[9]{\binom {n}{4}},\,$

For the 600-cell and 120-cell numbers:

$P_{120}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[357]{\binom {n+1}{4}}+[107]{\binom {n}{4}},\,$

$P_{600}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[1993]{\binom {n+1}{4}}+[543]{\binom {n}{4}},\,$

Schläfli-Poincaré (convex) polytope formula

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

For 4-dimensional (d = 4) regular convex polytopes:

${\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=N_{0}-N_{1}+N_{2}-N_{3}=V-E+F-C=0,\,$

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) of the regular convex polytope.

Recurrence equation

The recurrence equation seems to be the same for all polychoron numbers (TO BE VERIFIED for 5-cell, 16-cell and 8-cell numbers):

$P_{N_{0}}^{(4)}(n)=5P_{N_{0}}^{(4)}(n-1)-10P_{N_{0}}^{(4)}(n-2)+10P_{N_{0}}^{(4)}(n-3)-5P_{N_{0}}^{(4)}(n-4)+P_{N_{0}}^{(4)}(n-5)\,\,$

with initial conditions

$P_{N_{0}}^{(4)}(0)=?\,$

For the 5-cell numbers:

$P_{5}^{(4)}(n)=,\,$

For the 16-cell and 8-cell numbers:

$P_{8}^{(4)}(n)=,\,$

$P_{16}^{(4)}(n)=,\,$

For the 24-cell numbers:

$P_{24}^{(4)}(n)=5P_{24}^{(4)}(n-1)-10P_{24}^{(4)}(n-2)+10P_{24}^{(4)}(n-3)-5P_{24}^{(4)}(n-4)+P_{24}^{(4)}(n-5),\,$

For the 600-cell and 120-cell numbers:

$P_{120}^{(4)}(n)=5P_{120}^{(4)}(n-1)-10P_{120}^{(4)}(n-2)+10P_{120}^{(4)}(n-3)-5P_{120}^{(4)}(n-4)+P_{120}^{(4)}(n-5),\,$

$P_{600}^{(4)}(n)=5P_{600}^{(4)}(n-1)-10P_{600}^{(4)}(n-2)+10P_{600}^{(4)}(n-3)-5P_{600}^{(4)}(n-4)+P_{600}^{(4)}(n-5).\,$

Ordinary generating function

$G_{\{P_{N_{0}}^{(4)}(n)\}}(x)={\frac {x(1+[N_{0}-5]x+[???]x^{2}+[???]x^{3})}{(1-x)^{5}}}\,$

For the 5-cell numbers:

$G_{\{P_{5}^{(4)}(n)\}}(x)={\frac {x}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+0x^{3})}{(1-x)^{5}}},\,$

For the 16-cell and 8-cell numbers:

$G_{\{P_{8}^{(4)}(n)\}}(x)={\frac {x(1+3x+3x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,$

$G_{\{P_{16}^{(4)}(n)\}}(x)={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,$

For the 24-cell numbers:

$G_{\{P_{24}^{(4)}(n)\}}(x)={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+43x^{2}+9x^{3})}{(1-x)^{5}}},\,$

For the 600-cell and 120-cell numbers:

$G_{\{P_{120}^{(4)}(n)\}}(x)={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+357x^{2}+107x^{3})}{(1-x)^{5}}},\,$

$G_{\{P_{600}^{(4)}(n)\}}(x)={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+1993x^{2}+543x^{3})}{(1-x)^{5}}},\,$

Procedure to obtain the generating functions

$G_{\{P_{24}^{(4)}(n)\}}(x)=G_{\{n^{2}(3n^{2}-4n+2)\}}(x)=G_{\{3n^{4}-4n^{3}+2n^{2}\}}(x)=3G_{\{n^{4}\}}(x)-4G_{\{n^{3}\}}(x)+2G_{\{n^{2}\}}(x)\,$

$={\frac {x}{(1-x)^{5}}}\{3(1+11x+11x^{2}+x^{3})-4(1-x)(1+4x+x^{2})+2(1-x)^{2}(1+x)\}={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}},\,$

$G_{\{P_{120}^{(4)}(n)\}}(x)=G_{\{{\frac {n(145n^{3}-280n^{2}+179n-38)}{6}}\}}(x)={\frac {1}{6}}G_{\{145n^{4}-280n^{3}+179n^{2}-38n\}}(x)={\frac {1}{6}}\{145G_{\{n^{4}\}}(x)-280G_{\{n^{3}\}}(x)+179G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,$

$={\frac {x}{6(1-x)^{5}}}\{145(1+11x+11x^{2}+x^{3})-280(1-x)(1+4x+x^{2})+179(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}},\,$

$G_{\{P_{600}^{(4)}(n)\}}(x)=G_{\{{\frac {n(261n^{3}-504n^{2}+283n-38)}{2}}\}}(x)={\frac {1}{2}}G_{\{261n^{4}-504n^{3}+283n^{2}-38n\}}(x)={\frac {1}{2}}\{261G_{\{n^{4}\}}(x)-504G_{\{n^{3}\}}(x)+283G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,$

$={\frac {x}{2(1-x)^{5}}}\{261(1+11x+11x^{2}+x^{3})-504(1-x)(1+4x+x^{2})+283(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}},\,$

where (Cf. Generating function for regular orthotopic numbers)

$G_{\{n^{4}\}}(x)={\frac {x\ E_{4}(x)}{(1-x)^{5}}}={\frac {x\ (1+11x+11x^{2}+x^{3})}{(1-x)^{5}}},\,$

$G_{\{n^{3}\}}(x)={\frac {x\ E_{3}(x)}{(1-x)^{4}}}={\frac {x\ (1+4x+x^{2})}{(1-x)^{4}}},\,$

$G_{\{n^{2}\}}(x)={\frac {x\ E_{2}(x)}{(1-x)^{3}}}={\frac {x\ (1+x)}{(1-x)^{3}}},\,$

$G_{\{n\}}(x)={\frac {x\ E_{1}(x)}{(1-x)^{2}}}={\frac {x\ (1)}{(1-x)^{2}}}.\,$

Exponential generating function

$E_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,$

For the 5-cell numbers:

$E_{\{P_{5}^{(4)}(n)\}}(x)=,\,$

For the 16-cell and 8-cell numbers:

$E_{\{P_{8}^{(4)}(n)\}}(x)=,\,$

$E_{\{P_{16}^{(4)}(n)\}}(x)=(x+7x^{2}+6x^{3}+x^{4})\ e^{x},\,$

For the 24-cell numbers:

$E_{\{P_{24}^{(4)}(n)\}}(x)=,\,$

For the 600-cell and 120-cell numbers:

$E_{\{P_{120}^{(4)}(n)\}}(x)=,\,$

$E_{\{P_{600}^{(4)}(n)\}}(x)=,\,$

Dirichlet generating function

$D_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,$

For the 5-cell numbers:

$D_{\{P_{5}^{(4)}(n)\}}(x)=,\,$

For the 16-cell and 8-cell numbers:

$D_{\{P_{8}^{(4)}(n)\}}(x)=,\,$

$D_{\{P_{16}^{(4)}(n)\}}(x)=\zeta (s-4),\,$

For the 24-cell numbers:

$D_{\{P_{24}^{(4)}(n)\}}(x)=,\,$

For the 600-cell and 120-cell numbers:

$D_{\{P_{120}^{(4)}(n)\}}(x)=,\,$

$D_{\{P_{600}^{(4)}(n)\}}(x)=,\,$

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 $\scriptstyle A\,$ of nonnegative integers is called a basis of order $\scriptstyle g\,$ if $\scriptstyle g\,$ is the minimum number with the property that every nonnegative integer can be written as a sum of $\scriptstyle g\,$ elements in $\scriptstyle A\,$. 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 $\scriptstyle k\,$, i.e. every nonnegative integer can be written as a sum of $\scriptstyle k\,$ 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

$P_{N_{0}}^{(4)}(n)-P_{N_{0}}^{(4)}(n-1)=?,\,$

Partial sums

The partial sums correspond to 5-dimensional polychoron hyperpyramidal numbers.

$\sum _{n=1}^{m}P_{N_{0}}^{(4)}(n)=?,\,$

Partial sums of reciprocals

$\sum _{n=1}^{m}{\frac {1}{P_{N_{0}}^{(4)}(n)}}=?,\,$

Sum of reciprocals

$\sum _{n=1}^{\infty }{\frac {1}{P_{N_{0}}^{(4)}(n)}}=?,\,$

Table of formulae and values

N₀, N₁, N₂ and N₃ are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N₀ of vertices.

Regular polychoron numbers formulae and values

Rank r	N0	Name (N0, N1, N2, N3) Schläfli symbol	Formulae $P_{N_{0}}^{(4)}(n)=\,$ $\scriptstyle {\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[???]{\binom {n+1}{4}}+[???]{\binom {n}{4}},\,$	Generating function $G_{\{P_{N_{0}}^{(4)}(n)\}}(x)=\,$ $\scriptstyle {{x(1+??x+??x^{2}+??x^{3})} \over {(1-x)^{5}}}\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
0	5	Pentachoron 5 cell (5, 10, 10, 5) {3, 3, 3}	${\binom {n+3}{4}}$ ${\frac {n^{(4)}}{4!}}$ ${{(n+3)P_{4}^{(3)}(n)} \over 4}$	$x \over (1-x)^{5}\,$	0	1	5	15	35	70	126	210	330	495	715	1001	1365	A000332(n+3)
1	8	16 cell (8, 24, 32, 16) {3, 3, 4}	$\scriptstyle {\binom {n+3}{4}}+3{\binom {n+2}{4}}+3{\binom {n+1}{4}}+{\binom {n}{4}}\,$ ${n^{2}(n^{2}+2) \over 3}\,$	$\scriptstyle {\frac {x(1+3x+3x^{2}+x^{3})}{(1-x)^{5}}}\,$ ${\frac {x(1+x)^{3}}{(1-x)^{5}}}\,$	0	1	8	33	96	225	456	833	1408	2241	3400	4961	7008	A014820(n-1)
2	16	Tesseract 8 cell (16, 32, 24, 8) {4, 3, 3}	$\scriptstyle {\binom {n+3}{4}}+11{\binom {n+2}{4}}+11{\binom {n+1}{4}}+{\binom {n}{4}}\,$ $n^{4}\,$	$\scriptstyle {{x(1+11x+11x^{2}+x^{3})} \over {(1-x)^{5}}}\,$	0	1	16	81	256	625	1296	2401	4096	6561	10000	14641	20736	A000583
3	24	24 cell (24, 96, 96, 24) {3, 4, 3}	$\scriptstyle {\binom {n+3}{4}}+19{\binom {n+2}{4}}+43{\binom {n+1}{4}}+9{\binom {n}{4}}\,$ $n^{2}(3n^{2}-4n+2)\,$	$\scriptstyle {{x(1+19x+43x^{2}+9x^{3})} \over {(1-x)^{5}}}\,$	0	1	24	153	544	1425	3096	5929	10368	16929	26200	38841	55584	A092181
4	120	600 cell (120, 720, 1200, 600) {3, 3, 5}	$\scriptstyle {\binom {n+3}{4}}+115{\binom {n+2}{4}}+357{\binom {n+1}{4}}+107{\binom {n}{4}}\,$ $\scriptstyle {{n(145n^{3}-280n^{2}+179n-38)} \over 6}\,$	$\scriptstyle {{x(1+115x+357x^{2}+107x^{3})} \over {(1-x)^{5}}}\,$	0	1	120	947	3652	9985	22276	43435	76952	126897	197920	295251	424700	A092182
5	600	120 cell (600, 1200, 720, 120) {5, 3, 3}	$\scriptstyle {\binom {n+3}{4}}+595{\binom {n+2}{4}}+1993{\binom {n+1}{4}}+543{\binom {n}{4}}\,$ $\scriptstyle {{n(261n^{3}-504n^{2}+283n-38)} \over 2}\,$	$\scriptstyle {{x(1+595x+1993x^{2}+543x^{3})} \over {(1-x)^{5}}}\,$	0	1	600	4983	19468	53505	119676	233695	414408	683793	1066960	1592151	2290740	A092183

Table of related formulae and values

N₀, N₁, N₂ and N₃ are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N₀ of vertices.

Regular polychoron numbers related formulae and values

Rank r	N0	Name (N0, N1, N2, N3) Schläfli symbol	Order of basis $g_{\{P_{N_{0}}^{(4)}\}}=\,$ $N_{0}+?\,$	Differences $P_{N_{0}}^{(4)}(n)-\,$ $P_{N_{0}}^{(4)}(n-1)=\,$	Partial sums $\sum _{n=1}^{m}{P_{N_{0}}^{(4)}(n)}$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {P_{N_{0}}^{(4)}(n)}}$	Sum of Reciprocals $\sum _{n=1}^{\infty }{1 \over {P_{N_{0}}^{(4)}}}$
0	5 $\scriptstyle 2^{r+2}+0^{r}\,$	Pentachoron 5 cell (5, 10, 10, 5) {3, 3, 3}	$8?\,$ $(N_{0}+3?)\,$	$P_{3+1}^{(3)}(n)\,$ Tetrahedral numbers (A000292(n))	${\binom {m+4}{5}}\,$ ${\frac {m^{(5)}}{5!}}\,$ $\left(\!\!{\binom {m}{5}}\!\!\right)\,$ $\scriptstyle {\frac {m(m+1)(m+2)(m+3)(m+4)}{120}}\,$ A000389	$\,$	${\frac {4}{3}}$
1	8 $\scriptstyle 2^{r+2}+0^{r}\,$	16 cell (8, 24, 32, 16) {3, 3, 4}	$11?\,$ $(N_{0}+3?)\,$	${\,}_{c}P_{2\cdot 3}^{(3)}(n)\,$ Centered octahedral numbers (A001845(n-1))	${\binom {2m+2}{3}}\ {\frac {m^{2}+m+3}{20}}\,$ ${\frac {(2m)^{(3)}\ (m^{2}+m+3)}{5!}}\,$ $\scriptstyle {\frac {m(m+1)(2m+1)(m^{2}+m+3)}{30}}\,$ A061927(n-1)?	$\,$	$\,$
2	16 $\scriptstyle 2^{r+2}+0^{r}\,$	Tesseract 8 cell (16, 32, 24, 8) {4, 3, 3}	$19\,$ $(N_{0}+3)\,$	${\,}_{c}P_{2^{3}}^{(3)}(n)+6\ Y_{5}^{(3)}(n-1)\,$ Centered cubic(n) + 6 Square pyramidal(n-1) Rhombic dodecahedral numbers (A005917(n-1))	${\binom {2m+2}{3}}\ {\frac {3m^{2}+3m-1}{20}}\,$ $\scriptstyle {\frac {(2m)^{(3)}\ (3m^{2}+3m-1)}{5!}}\,$ $\scriptstyle {\frac {m(m+1)(2m+1)(3m^{2}+3m-1)}{30}}\,$ A000538	$\,$	$\zeta (4)={{\pi ^{4}} \over 90}\,$
3	24 $\scriptstyle (r+1)!\,$ $\scriptstyle (r+1)\cdot r!\,$	24 cell (24, 96, 96, 24) {3, 4, 3}	$28?\,$ $(N_{0}+4?)\,$	$\scriptstyle 12n^{3}-30n^{2}+28n-9\,$	$\scriptstyle {\binom {m+1}{2}}\ {\frac {18m^{3}-3m^{2}-7m+7}{15}}\,$ $\scriptstyle {\frac {m(m+1)(18m^{3}-3m^{2}-7m+7)}{30}}\,$	$\,$	$\,$
4	120 $\scriptstyle (r+1)!\,$ $\scriptstyle (r+1)\cdot r!\,$	600 cell (120, 720, 1200, 600) {3, 3, 5}	$125?\,$ $(N_{0}+5?)\,$	$\scriptstyle {\frac {290n^{3}-855n^{2}+889n-321}{3}}\,$	$\scriptstyle {{\binom {m+1}{2}}\ {\frac {58m^{3}-53m^{2}-11m+12}{6}}}\,$ $\scriptstyle {\frac {m(m+1)(58m^{3}-53m^{2}-11m+12)}{12}}\,$	$\,$	$\,$
5	600 $\scriptstyle r\cdot r!\,$	120 cell (600, 1200, 720, 120) {5, 3, 3}	$606?\,$ $(N_{0}+6?)\,$	$\scriptstyle {522n^{3}-1539n^{2}+1561n-543}\,$	$\scriptstyle {{\binom {m+1}{2}}\ {\frac {1566m^{3}-1431m^{2}-689m+584}{30}}}\,$ $\scriptstyle {\frac {m(m+1)(1566m^{3}-1431m^{2}-689m+584)}{60}}\,$	$\,$	$\,$

Table of sequences

Regular polychoron numbers sequences

N0	$P_{N_{0}}^{(4)}(n),\ n\geq 0$ sequences
5	{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, ...}
8	{0, 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, 7008, 9633, 12936, 17025, 22016, 28033, 35208, 43681, 53600, 65121, 78408, 93633, 110976, 130625, 152776, 177633, ...}
16	{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, ...}
24	{0, 1, 24, 153, 544, 1425, 3096, 5929, 10368, 16929, 26200, 38841, 55584, 77233, 104664, 138825, 180736, 231489, 292248, 364249, 448800, 547281, 661144, 791913, 941184, ...}
120	{0, 1, 120, 947, 3652, 9985, 22276, 43435, 76952, 126897, 197920, 295251, 424700, 592657, 806092, 1072555, 1400176, 1797665, 2274312, 2839987, 3505140, 4280801, ...}
600	{0, 1, 600, 4983, 19468, 53505, 119676, 233695, 414408, 683793, 1066960, 1592151, 2290740, 3197233, 4349268, 5787615, 7556176, 9701985, 12275208, 15329143, 18920220, ...}

See also

Centered regular polychoron 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.