Square hyperpyramidal numbers

The simplicial polytopic numbers are triangular (hyper)pyramidal numbers, starting with a triangle for dimension d = 2, then do a pyramidal stacking for each extra dimension.

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

While the simplicial polytopic numbers correspond to regular polytopes (regular simplex polytopes), the square (hyper)pyramidal numbers, for d ≥ 3, correspond to nonregular polytopes (square (hyper)pyramids.) They are nonetheless especially interesting since they are the building blocks for the orthoplicial polytopic numbers, which are regular polytopes (regular othoplex polytopes.)

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

1 (2,1)-Pascal triangle or (1,2)-Pascal triangle and square (hyper)pyramidal numbers
2 Formulae
3 Descartes-Euler (convex) polytope formula
4 Recurrence equation
5 Generating function
6 Order of basis
7 Differences
8 Partial sums
9 Partial sums of reciprocals
10 Sum of reciprocals
11 Table of formulae and values
12 Table of related formulae and values
13 Table of sequences
14 See also
15 Notes
16 External links

(2,1)-Pascal triangle or (1,2)-Pascal triangle and square (hyper)pyramidal numbers

(2,1)-Pascal (rectangular) triangle columns and square (hyper)pyramidal numbers

(1,2)-Pascal (rectangular) triangle falling diagonals and square (hyper)pyramidal numbers

Formulae

Y^{(d)}(d+2,n)=?\,

Descartes-Euler (convex) polytope formula

Recurrence equation

Y^{(d)}(d+2,n)=?,\,

Y^{(d)}(d+2,0)=0,\ Y^{(d)}(d+2,1)=1,\ Y^{(d)}(d+2,2)=?.\,

Generating function

G_{Y^{(d)}}(d+2,x)=?\,

Order of basis

Differences

Y^{(d)}(d+2,n)-Y^{(d)}(d+2,n-1)=?\,

Partial sums

\sum _{n=1}^{m}Y^{(d)}(d+2,n)=?\,

Partial sums of reciprocals

\sum _{n=1}^{m}{\frac {1}{Y^{(d)}(d+2,n)}}=?\,

Sum of reciprocals

\sum _{n=1}^{\infty }{\frac {1}{Y^{(d)}(d+2,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 square hyperpyramidal numbers are listed by increasing number N₀ of vertices.

**Square hyperpyramidal numbers formulae and values**
d	(N₀, N₁, N₂, ...) Schläfli symbol^[1]	Formulae $Y_{d+2}^{(d)}(n)=\,$ ${\binom {n+d-2}{d-1}}{\frac {(2n+d-2)}{d}}\,$	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
1	Square gnomon (2) {}	${\binom {n-1}{0}}{\frac {(2n-1)}{1}}\,$ $2n-1+0^{n}\,$	1	3	5	7	9	11	13	15	17	19	21	23	A004273(n)
2	Square (3, 3) {3}	${\binom {n}{1}}{\frac {(2n)}{2}}\,$ $n^{2}\,$	1	4	9	16	25	36	49	64	81	100	121	144	A000290(n)
3	Square pyramidal (, , ) {, }	${\binom {n+1}{2}}{\frac {(2n+1)}{3}}\,$ ${\frac {1}{4}}{\binom {2n+2}{3}}\,$ ${\frac {(2n)^{(3)}}{4!}}\,$ ^[2] ${n(n+1)(2n+1)} \over 6\,$ $T_{n}\ {\frac {(2n+1)}{3}}\,$	1	5	14	30	55	91	140	204	285	385	506	650	A000330(n)
4	(, , , ) {, , }	${\binom {n+2}{3}}{\frac {(2n+2)}{4}}\,$	1	6	20	50	105	196	336	540	825	1210	1716	2366	A002415(n+1)
5	(, , , , ) {, , , }	${\binom {n+3}{4}}{\frac {(2n+3)}{5}}\,$	1	7	27	77	182	378	714	1254	2079	3289	5005	7371	A005585(n)
6	(, , , , , ) {, , , , }	${\binom {n+4}{5}}{\frac {(2n+4)}{6}}\,$	1	8	35	112	294	672	1386	2640	4719	8008	13013	20384	A040977(n-1)
7	(, , , , , , ) {, , , , , }	${\binom {n+5}{6}}{\frac {(2n+5)}{7}}\,$	1	9	44	156	450	1122	2508	5148	9867	17875	30888	51272	A050486(n-1)
8	(, , , , , , , ) {, , , , , , }	${\binom {n+6}{7}}{\frac {(2n+6)}{8}}\,$	1	10	54	210	660	1782	4290	9438	19305	37180	68068	119340	A053347(n-1)
9	(, , , , , , , , ) {, , , , , , , }	${\binom {n+7}{8}}{\frac {(2n+7)}{9}}\,$	1	11	65	275	935	2717	7007	16445	35750	72930	140998	260338	A054333(n-1)
10	(, , , , , , , , , ) {, , , , , , , , }	${\binom {n+8}{9}}{\frac {(2n+8)}{10}}\,$	1	12	77	352	1287	4004	11011	27456	63206	136136	277134	537472	A054334(n-1)
11	(, , , , , , , , , , ) {, , , , , , , , , }	${\binom {n+9}{10}}{\frac {(2n+9)}{11}}\,$	1	13	90	442	1729	5733	16744	44200	107406	243542	520676	1058148	A057788(n-1)
12	(, , , , , , , , , , , ) {, , , , , , , , , , }	${\binom {n+10}{11}}{\frac {(2n+10)}{12}}\,$	1	14	104	546	2275	8008	24752	68952	176358	419900	940576	1998724	Axxxxxx

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 square hyperpyramidal numbers are listed by increasing number N₀ of vertices.

**Square hyperpyramidal numbers related formulae and values**
d	Generating function $G_{Y^{(d)}(d+2)}(x)=\,$ ${\frac {x(1+x)}{(1-x)^{d+1}}}\,$	Order of basis ^[3]^[4]^[5]	Differences $Y^{(d)}(d+2,n)-\,$ $Y^{(d)}(d+2,n-1)=\,$ $Y^{(d-1)}(d+1,n)\,$	Partial sums $\sum _{n=1}^{m}{Y^{(d)}(d+2,n)}=$ $Y^{(d+1)}(d+3,m)\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {Y^{(d)}(d+2,n)}}=$	Sum of reciprocals^[6] $\sum _{n=1}^{\infty }{1 \over {Y^{(d)}(d+2,n)}}=$
1	${\frac {x(1+x)}{(1-x)^{2}}}\,$	$2\,$	$2-0^{n-1}\,$	$Y^{(2)}(4,m)\,$	$\,$	$\,$
2	${\frac {x(1+x)}{(1-x)^{3}}}\,$	$4\,$	$2n-1\,$	$Y^{(3)}(5,m)\,$	$\,$	$\,$
3	${\frac {x(1+x)}{(1-x)^{4}}}\,$	$\,$	${\frac {2n^{2}}{2!}}\,$ $n^{2}\,$	$Y^{(4)}(6,m)\,$	$\,$	$\,$
4	${\frac {x(1+x)}{(1-x)^{5}}}\,$	$\,$	${\frac {n(1+2n)(1+n)}{3!}}\,$	$Y^{(5)}(7,m)\,$	$\,$	$\,$
5	${\frac {x(1+x)}{(1-x)^{6}}}\,$	$\,$	${\frac {2n(1+n)^{2}(2+n)}{4!}}\,$	$Y^{(6)}(8,m)\,$	$\,$	$\,$
6	${\frac {x(1+x)}{(1-x)^{7}}}\,$	$\,$	${\frac {n(18+45n+40n^{2}+15n^{3}+2n^{4})}{5!}}\,$	$Y^{(7)}(9,m)\,$	$\,$	$\,$
7	${\frac {x(1+x)}{(1-x)^{8}}}\,$	$\,$	$\,$	$Y^{(8)}(10,m)\,$	$\,$	$\,$
8	${\frac {x(1+x)}{(1-x)^{9}}}\,$	$\,$	$\,$	$Y^{(9)}(11,m)\,$	$\,$	$\,$
9	${\frac {x(1+x)}{(1-x)^{10}}}\,$	$\,$	$\,$	$Y^{(10)}(12,m)\,$	$\,$	$\,$
10	${\frac {x(1+x)}{(1-x)^{11}}}\,$	$\,$	$\,$	$Y^{(11)}(13,m)\,$	$\,$	$\,$
11	${\frac {x(1+x)}{(1-x)^{12}}}\,$	$\,$	$\,$	$Y^{(12)}(14,m)\,$	$\,$	$\,$
12	${\frac {x(1+x)}{(1-x)^{13}}}\,$	$\,$	$\,$	$Y^{(13)}(15,m)\,$	$\,$	$\,$

Table of sequences

**Square hyperpyramidal numbers sequences**
d	Sequence
1	{0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, ...}
2	{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, ...}
3	{0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, ...}
4	{0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, ...}
5	{0, 1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, 95634, 118910, 146510, 179010, 217035, 261261, 312417, 371287, 438712, ...}
6	{0, 1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, 311696, 407330, 526240, 672750, 851760, 1068795, 1330056, 1642473, ... }
7	{0, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875, 30888, 51272, 82212, 127908, 193800, 286824, 415701, 591261, 826804, 1138500, 1545830, 2072070, 2744820, 3596580, 4665375, 5995431, ...}
8	{0, 1, 10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068, 119340, 201552, 329460, 523260, 810084, 1225785, 1817046, 2643850, 3782350, 5328180, 7400250, 10145070, 13741650, 18407025, ...}
9	{0, 1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930, 140998, 260338, 461890, 791350, 1314610, 2124694, 3350479, 5167525, 7811375, 11593725, 16921905, 24322155, 34467225, 48208875, ...}
10	{0, 1, 12, 77, 352, 1287, 4004, 11011, 27456, 63206, 136136, 277134, 537472, 999362, 1790712, 3105322, 5230016, 8580495, 13748020, 21559395, 33153120, 50075025, 74397180, 108864405, ...}
11	{0, 1, 13, 90, 442, 1729, 5733, 16744, 44200, 107406, 243542, 520676, 1058148, 2057510, 3848222, 6953544, 12183560, 20764055, 34512075, 56071470, 89224590, 139299615, 213696795, ...}
12	{0, 1, 14, 104, 546, 2275, 8008, 24752, 68952, 176358, 419900, 940576, 1998724, 4056234, 7904456, 14858000, 27041560, 47805615, 82317690, 138389160, 227613750, 366913365, 580610160, ...}

Notes

↑ Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
↑ HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
↑ Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.

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.

[Schlafli-1] Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.

[RisingFactorial-2] Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.

[FermatsPolygonalNumberTheorem-3] Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.

[ON_REGULAR_POLYTOPE_NUMBERS-4] HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.

[5] Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.

[6] Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.

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Square hyperpyramidal numbers

Contents

(2,1)-Pascal triangle or (1,2)-Pascal triangle and square (hyper)pyramidal numbers

Formulae

Descartes-Euler (convex) polytope formula

Recurrence equation

Generating function

Order of basis

Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals

Table of formulae and values

Table of related formulae and values

Table of sequences

See also

Notes

External links

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