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Square hyperpyramidal numbers

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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

Contents

(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)}_{d+2}(n) =\,


\binom{n+d-2}{d-1} \frac{(2n+d-2)}{d}\,

n = 0 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\,

0 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\,

0 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)}\over6\,


T_n\ \frac{(2n+1)}{3}\,

0 1 5 14 30 55 91 140 204 285 385 506 650 A000330(n)
4

(, , , )

{, , }

\binom{n+2}{3} \frac{(2n+2)}{4}\, 0 1 6 20 50 105 196 336 540 825 1210 1716 2366 A002415(n+1)
5

(, , , , )

{, , , }

\binom{n+3}{4} \frac{(2n+3)}{5}\, 0 1 7 27 77 182 378 714 1254 2079 3289 5005 7371 A005585(n)
6

(, , , , , )

{, , , , }

\binom{n+4}{5} \frac{(2n+4)}{6}\, 0 1 8 35 112 294 672 1386 2640 4719 8008 13013 20384 A040977(n-1)
7

(, , , , , , )

{, , , , , }

\binom{n+5}{6} \frac{(2n+5)}{7}\, 0 1 9 44 156 450 1122 2508 5148 9867 17875 30888 51272 A050486(n-1)
8

(, , , , , , , )

{, , , , , , }

\binom{n+6}{7} \frac{(2n+6)}{8}\, 0 1 10 54 210 660 1782 4290 9438 19305 37180 68068 119340 A053347(n-1)
9

(, , , , , , , , )

{, , , , , , , }

\binom{n+7}{8} \frac{(2n+7)}{9}\, 0 1 11 65 275 935 2717 7007 16445 35750 72930 140998 260338 A054333(n-1)
10

(, , , , , , , , , )

{, , , , , , , , }

\binom{n+8}{9} \frac{(2n+8)}{10}\, 0 1 12 77 352 1287 4004 11011 27456 63206 136136 277134 537472 A054334(n-1)
11

(, , , , , , , , , , )

{, , , , , , , , , }

\binom{n+9}{10} \frac{(2n+9)}{11}\, 0 1 13 90 442 1729 5733 16744 44200 107406 243542 520676 1058148 A057788(n-1)
12

(, , , , , , , , , , , )

{, , , , , , , , , , }

\binom{n+10}{11} \frac{(2n+10)}{12}\, 0 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 + 45 n + 40 n^2 + 15 n^3 + 2 n^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, ...}


See also

Centered square hyperpyramidal numbers, i.e. (Centered squares) hyperpyramidal numbers.

Notes

  1. Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  2. Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
  3. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  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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