A098359 -id:A098359 - cp's OEIS frontend

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Marc LeBrun

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.
Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001
A permutation of A061017. - Matthew Vandermast, Feb 28 2003
In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003
Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003
Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005
Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element  = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012
Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013
T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019
From Eric Lengyel, Jun 28 2023: (Start)
X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.
X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)
T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

			The array T starts in row n=1 with columns m>=1 as:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30
   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45
   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60
   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75
   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90
   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105
   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120
   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135
  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150
The triangle X(n, m) begins
   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:   1
   2:   2  2
   3:   3  4  3
   4:   4  6  6  4
   5:   5  8  9  8  5
   6:   6 10 12 12 10  6
   7:   7 12 15 16 15 12  7
   8:   8 14 18 20 20 18 14  8
   9:   9 16 21 24 25 24 21 16  9
  10:  10 18 24 28 30 30 28 24 18 10
  11:  11 20 27 32 35 36 35 32 27 20 11
  12:  12 22 30 36 40 42 42 40 36 30 22 12
  13:  13 24 33 40 45 48 49 48 45 40 33 24 13
  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14
  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15
  ... Formatted by _Wolfdieter Lang_, Dec 02 2014

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 5-6.

T. D. Noe, Rows n = 1..100 of triangle, flattened
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
G. W. Leibniz, Dissertatio de arte combinatoria, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.
Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.
Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

/* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jul 12 2019

seq(seq(i*(n-i),i=1..n-1),n=2..10); # Robert Israel, Dec 14 2015

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
f[n_] := Table[SeriesCoefficient[E^(x + y) (1+ x - y +x*y-y^2), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11,0]] (* Stefano Spezia, Jul 12 2019 *)

PARI
```
A003991(n,k) = if(k<1 || n<1,0,k*n)
```

Rectangular array: T(n, m) = n*m, n>=1, m>= 1.
Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.
Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004
G.f. as rectangular array: x*y/((1 - x)^2*(1 - y)^2).
a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007
As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007
As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012
G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015
E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - Stefano Spezia, Jul 12 2019
a(n) = (1/2)*t + (n - 1/4)*t^2 - (1/4)*t^4 - n^2 + n, where t = floor(sqrt(2*n) + 1/2). - Ridouane Oudra, Nov 21 2020
a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - Antti Karttunen, Dec 13 2021
T(n,k) = A002620(n+k) - A002620(n-k). - Michel Marcus, Jan 06 2023
T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y < z. - Clark Kimberling, Jan 22 2024
E.g.f. as rectangular array: x*y*exp(x+y). - Stefano Spezia, Jun 27 2025

More terms from Michael Somos

A098358 Multiplication table of the triangular numbers read by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 9, 6, 10, 18, 18, 10, 15, 30, 36, 30, 15, 21, 45, 60, 60, 45, 21, 28, 63, 90, 100, 90, 63, 28, 36, 84, 126, 150, 150, 126, 84, 36, 45, 108, 168, 210, 225, 210, 168, 108, 45, 55, 135, 216, 280, 315, 315, 280, 216, 135, 55, 66, 165, 270, 360, 420, 441, 420, 360

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

The number of rectangles to be found in a grid of size X by y. For example a(2, 2) = 9 since a 2 x 2 grid contains one rectangle of size 2 X 2, 4 of size 1 X 2 and 4 of size 1 X 1. - Hugo van der Sanden, May 24 2007

			Triangle begins:
   1;
   3,  3;
   6,  9,  6;
  10, 18, 18, 10;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A003991, A098359, A098360, A098361.

a[n_, k_] := Binomial[k + 1, 2]*Binomial[n + 1, 2]; Table[a[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

a(m,n) = m*(m+1)*n*(n+1)/4.

G.f.: x*y / ((1-x)^3 * (1-y)^3). - Ralf Stephan, Oct 27 2004

A098361 Multiplication table of the factorial numbers read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 24, 120, 24, 12, 12, 24, 120, 720, 120, 48, 36, 48, 120, 720, 5040, 720, 240, 144, 144, 240, 720, 5040, 40320, 5040, 1440, 720, 576, 720, 1440, 5040, 40320, 362880, 40320, 10080, 4320, 2880, 2880, 4320, 10080, 40320, 362880

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - Stephen Crowley, Mar 28 2007
Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - Roger L. Bagula, Oct 29 2008
The triangle T(n, k) = k!*(n-k)! appears as denominators in the coefficients of the Niven polynomials x^n*(1 - x)^n/n! = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!). These polynomials are used in a proof that Pi^2 (hence Pi) is irrational. See the Niven and Havil references. - Wolfdieter Lang, May 07 2018; corrected by Dimitri Papadopoulos, Nov 30 2023
The case T(n+1,k) = k!*(n-k+1)!, 1 <= k <= n+1, n >= 0 is the number of choices for forming a cluster (compact group) of k numbered items arranged in a line on a set of permutations of n numbered items arranged in a line. - Igor Victorovich Statsenko, Oct 13 2023
The numbers T(n,k) also appear in the denominators of the partial fraction expansion of 1/(x*(x+1)*...*(x+n)) = Sum_{k=0..n} (-1)^k * 1/(T(n,k)*(x+k)). - Dimitri Papadopoulos, Nov 30 2023
It follows from the previous comment that the numbers T(n,k) also appear in the denominators of the coefficients of the logarithms of the integral of 1/(x*(x+1)*...*(x+n)): c + Sum{k=0...n} (-1)^k * 1/(T(n,k)) * ln(x+k). - Colin Linzer, Dec 18 2024

			The array A(n, k) starts in row n=0 with columns k >= 0 as:
       1,      1,      2,       6,      24,      120, ...
       1,      1,      2,       6,      24,      120, ...
       2,      2,      4,      12,      48,      240, ...
       6,      6,     12,      36,     144,      720, ...
      24,     24,     48,     144,     576,     2880, ...
     120,    120,    240,     720,    2880,    14400, ...
     720,    720,   1440,    4320,   17280,    86400, ...
    5040,   5040,  10080,   30240,  120960,   604800, ...
   40320,  40320,  80640,  241920,  967680,  4838400, ...
  362880, 362880, 725760, 2177280, 8709120, 43545600, ...
  ...
The triangle T(n, k) begins:
n\k       0      1     2     3     4     5     6     7     8      9      10...
0:        1
1:        1      1
2:        2      1     2
3:        6      2     2     6
4:       24      6     4     6    24
5:      120     24    12    12    24   120
6:      720    120    48    36    48   120   720
7:     5040    720   240   144   144   240   720  5040
8:    40320   5040  1440   720   576   720  1440  5040 40320
9:   362880  40320 10080  4320  2880  2880  4320 10080 40320 362880
10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800
... - _Wolfdieter Lang_, May 07 2018

R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

Stefano Spezia, First 101 antidiagonals of the array, flattened
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
I. V. Statsenko, Problem on variants of cluster formation at permutations in ordered structures, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2023, pp. 7-10. In Russian.

Row sums A003149.

Cf. A003991, A098358, A098359, A098360.

F:=Factorial; [F(n-k)*F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2022

seq(print(seq(k!*(n-k)!,k=0..n)),n=0..6); # Peter Luschny, Aug 23 2010

Table[(n+1)!*Beta[n-k+1, k+1], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Oct 29 2008 *)

f=factorial; flatten([[f(n-k)*f(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 12 2022

T(n, k) = k!*(n-k)! =  n!/C(n,k), (0<=k<=n). - Peter Luschny, Aug 23 2010
Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - R. J. Mathar, Dec 10 2010
E.g.f. as array: 1/((1 - x)*(1 - y)). - Stefano Spezia, Jul 10 2020

A098360 Multiplication table of the cube numbers read by antidiagonals.

Original entry on oeis.org

1, 8, 8, 27, 64, 27, 64, 216, 216, 64, 125, 512, 729, 512, 125, 216, 1000, 1728, 1728, 1000, 216, 343, 1728, 3375, 4096, 3375, 1728, 343, 512, 2744, 5832, 8000, 8000, 5832, 2744, 512, 729, 4096, 9261, 13824, 15625, 13824, 9261, 4096, 729, 1000, 5832, 13824

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

			1; 8,8; 27,64,27; 64,216,216,64; ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A003991, A098358, A098359, A098361.

Row sums: A145216. - N. J. A. Sloane, May 31 2009

Flat(List([2..11],m->List([1..m-1],i->i^3*(m-i)^3))); # Muniru A Asiru, Jun 27 2018

seq(seq(i^3*(m-i)^3,i=1..m-1),m=2..10); # Robert Israel, Jun 27 2018

With[{s = Range[10]^3}, Table[s[[#]] s[[j]] &[i - j + 1], {i, Length@s}, {j, i}]] // Flatten (* Michael De Vlieger, Jun 27 2018 *)

G.f. as rectangular array: [xy(1+4x+x^2)(1+4y+y^2)] / [(1-x)^4 * (1-y)^4 ]. - Ralf Stephan, Oct 27 2004, corrected by Robert Israel, Jun 27 2018

a(n) = A003991(n)^3.- Robert Israel, Jun 27 2018

More terms from Ralf Stephan, Oct 27 2004

Offset corrected by Robert Israel, Jun 27 2018

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A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

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A098358 Multiplication table of the triangular numbers read by antidiagonals.

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A098361 Multiplication table of the factorial numbers read by antidiagonals.

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A098360 Multiplication table of the cube numbers read by antidiagonals.

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