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User:Peter Luschny/FigurateNumber
Figurate number — a very short introduction
KEYWORDS: Polygonal number, orthotopic number, Eulerian polynomials
k\n  0  1  2  3  4  5  6  
5  0  −1  5  18  38  65  99  
4  0  −1  4  15  32  55  84  
3  0  −1  3  12  26  45  69  
2  0  −1  2  9  20  35  54  A014107 
1  0  −1  1  6  14  25  39  A095794 
0  0  1  0  −3  −8  −15  −24  A005563 A013648 A067998 A082562 A131386 A132411 
1  0  1  1  0  −2  −5  −9  A000096 A080956 A132337 A161680 A175631 
2  0  1  2  3  4  5  6  A001477 
3  0  1  3  6  10  15  21  A000217, A089594 
4  0  1  4  9  16  25  36  A000290 A162395 
5  0  1  5  12  22  35  51  A000326 
Polygonal number
The history of the figurate numbers goes back to the use of counting boards where small stones were used to represent numbers. Obviously it was helpful to arrange the stones in a figurative way, to make little pictograms for numbers.
Citing from Karl Menninger, "Number words and number symbols":
The Greeks called the counting board or table abakion, ...,
the counters were called psephoi (pebbles).

The triangular numbers are defined for n = 1,2,3,.. as
the square numbers as
the pentagonal numbers as
In general, the polygonal numbers are defined as
The sequence of partial sums of (k fixed) are the pyramidal numbers. These numbers can be represented as spatial figurations.
Polygonal numbers are arithmetic sequences of order 2, pyramidal numbers are arithmetic sequences of order 3, ... In general, a figurate number of dimension r is a member of an arithmetic sequence of order r.
The roman geometers Epaphroditus and Vitrius Rufus (circa 150 AD) found the pyramidal number formula

Fermat claimed to have a proof for the following proposition, which is a generalization of a theorem of Lagrange and of some interest in number theory, the polygonal number theorem:
Assume k > 2. Every integer n > 0 can be written as a sum of at most k polygonal numbers. 
The theorem was proved by Legendre and Cauchy.
10 is the fourth triangular number A000217.
16 is the fourth square number A000290.
22 is the fourth pentagonal number A000326.
28 is the fourth hexagonal number A000384.
19 is the fourth centered triangular number A005448.
25 is the fourth centered square number A001844.
31 is the fourth centered pentagonal number A005891.
37 is the fourth centered hexagonal number A003215.
Author of the plots: Stefan Friedrich Birkner, License: Creative Commons AttributionShareAlike 3.0 Unported.
A recommended introduction to polygonal numbers is: Dan MacKinnon, Triangulating Polygonal Numbers, Mathematics Teacher, Vol. 103, No. 7, March 2010.
Orthotopic number
The generating function of the regular orthotopic numbers is given essentially by the Eulerian polynomials. This was established by Euler in 1749.
To disambiguate: Orthotopic can mean in the normal or usual position. For instance it is used in clinical medicine in this sens. Another meaning is derived from parallelotope which is called an orthotope if the spanning vectors are mutually perpendicular. In this article it is used in the latter sens.
With Maple the Eulerian Polynomials (as defined by D. E. Knuth and now also by NIST Digital Library of Mathematical Functions) can be computed as:
A := proc(n,x) if n = 0 then 1 else x*(1x)*diff(A(n1,x),x)+A(n1,x)*(1+(n1)*x) fi end: seq(print(sort(expand(A(n,x)))),n=0..5); 1 1 x + 1 x^2 + 4 x + 1 x^3 + 11 x^2 + 11 x + 1 x^4 + 26 x^3 + 66 x^2 + 26 x + 1
The generating function of the regular orthotopic numbers is given by G(n,x) = xA(n, x)/(1−x)^{n+1}. Looking at the unshifted version F(n,x) = A(n, x)/(1−x)^{n+1} we find:
F := proc(n,x) A(n,x)/(1x)^(n+1) end: To compute the table below: seq(print([n^k],sort(factor(F(k,x))), seq(coeff(series(F(k,x),x,12),x,j),j=0..5)),k=0..5); [n^0]: (1x)^(1), 1, 1, 1, 1, 1, 1 [n^1]: (1x)^(2), 1, 2, 3, 4, 5, 6 [n^2]: (1x)^(3)(x+1), 1, 4, 9, 16, 25, 36 [n^3]: (1x)^(4)(x^2+4x+1), 1, 8, 27, 64, 125, 216 [n^4]: (1x)^(5)(x+1)(x^2+10x+1), 1,16, 81, 256, 625,1296 [n^5]: (1x)^(6)(x^4+26x^3+66x^2+26x+1), 1,32,243,1024,3125,7776
Reading the rectangular array by diagonals in western style (left to right) gives the
Triangle of orthotopic numbers A009998  
1  
1  1  
1  2  1  
1  4  3  1  
1  8  9  4  1  
1  16  27  16  5  1  
1  32  81  64  25  6  1 
Peter Bala observed that this is the Hilbert transform of the Eulerian numbers A123125. (See A145905 for the definition of the Hilbert transform of a triangular array.) Note that this is just a rephrasing of Euler's 1749 theorem. Unfortunately several different conventions with regard to the enumeration of the Eulerian numbers are in use. Bala remarks: 'The triangle of Eulerian numbers A008292 [can be viewed] as the coefficients of hpolynomials of ndimensional permutohedra of type A).' (See also the literature cited by Bala in A145905.)
The generating function of the centered regular orthotopic numbers is given by
H := proc(n,x) (1+x)*A(n,x)/(1x)^(n+1) end: To compute the table below: seq(print([n^k],sort(factor(H(k,x))), seq(coeff(series(H(k,x),x,12),x,j),j=0..5)),k=0..5); [n^0]: (1x)^(1)(1+x), 1, 2, 2, 2, 2, 2 [n^1]: (1x)^(2)(1+x), 1, 3, 5, 7, 9, 11 [n^2]: (1x)^(3)(1+x)(x+1), 1, 5, 13, 25, 41, 61 [n^3]: (1x)^(4)(1+x)(x^2+4x+1), 1, 9, 35, 91, 189, 341 [n^4]: (1x)^(5)(1+x)(x+1)(x^2+10x+1), 1,17, 97, 337, 881, 1921 [n^5]: (1x)^(6)(1+x)(x^4+26x^3+66x^2+26x+1), 1,33,275,1267,4149,10901
Reading the rectangular array by diagonals in western style (left to right) gives the
Triangle of centered orthotopic numbers A179927  
1  
1  2  
1  3  2  
1  5  5  2  
1  9  13  7  2  
1  17  35  25  9  2  
1  33  97  91  41  11  2 
Note that this triangle is related to the triangle A008518. In a more formal setup the exact wording depends on the definition of the Eulerian polynomials. Row sums are in A179928.
A linkfarm to the database corresponding to the rows of the two numerical rectangles above. (Note that the offset of the sequences possibly needs to be adjusted to meet your conventions.)
GF  0  1  2  3  4  5 
A(n,x)x/(1x)^(n+1)  A057427  A001477  A000290  A000578  A000583  A000584 
A(n,x)(x+1)/(1x)^(n+1)  A040000  A005408  A001844  A005898  A008514  A008515 
See the impressive contribution of Daniel Forgues to the Classifications of figurate numbers. A recommended reading is Regular polytopes by H. S. M. Coxeter, Dover Publications.