%I #184 Jul 03 2024 14:33:45

%S 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,

%T 14,18,20,20,18,14,8,9,16,21,24,25,24,21,16,9,10,18,24,28,30,30,28,24,

%U 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

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

%C 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.

%C 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

%C A permutation of A061017. - _Matthew Vandermast_, Feb 28 2003

%C 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

%C Permanent of upper right n X n corner is A000442. - _Marc LeBrun_, Dec 11 2003

%C 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

%C 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 <m+1|S_+|m> = 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 |<m+1|S_+|m><m|S_-|m+1>| 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

%C Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - _Charlie Marion_, Apr 22 2013

%C 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

%C From _Eric Lengyel_, Jun 28 2023: (Start)

%C 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.

%C 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)

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.

%H T. D. Noe, <a href="/A003991/b003991.txt">Rows n = 1..100 of triangle, flattened</a>

%H Iva Kodrnja and Helena Koncul, <a href="https://arxiv.org/abs/2405.10747">Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N))</a>, arXiv:2405.10747 [math.NT], 2024. See p. 10.

%H G. W. Leibniz, <a href="/A003991/a003991.pdf">Dissertatio de arte combinatoria</a>, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).

%H Abdelkader Necer, <a href="http://dx.doi.org/10.5802/jtnb.205">Séries formelles et produit de Hadamard</a>, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.

%F Rectangular array: T(n, m) = n*m, n>=1, m>= 1.

%F Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.

%F 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

%F G.f. as rectangular array: x * y / [ (1-x)^2 * (1-y)^2 ].

%F a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - _Hieronymus Fischer_, Aug 08 2007

%F 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

%F 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

%F 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

%F E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - _Stefano Spezia_, Jul 12 2019

%F 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

%F a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - _Antti Karttunen_, Dec 13 2021

%F T(n,k) = A002620(n+k) - A002620(n-k). - _Michel Marcus_, Jan 06 2023

%F 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 The array T starts in row n=1 with columns m>=1 as:

%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

%e 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

%e 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

%e 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

%e 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

%e 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

%e 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

%e 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120

%e 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135

%e 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

%e The triangle X(n, m) begins

%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

%e 1: 1

%e 2: 2 2

%e 3: 3 4 3

%e 4: 4 6 6 4

%e 5: 5 8 9 8 5

%e 6: 6 10 12 12 10 6

%e 7: 7 12 15 16 15 12 7

%e 8: 8 14 18 20 20 18 14 8

%e 9: 9 16 21 24 25 24 21 16 9

%e 10: 10 18 24 28 30 30 28 24 18 10

%e 11: 11 20 27 32 35 36 35 32 27 20 11

%e 12: 12 22 30 36 40 42 42 40 36 30 22 12

%e 13: 13 24 33 40 45 48 49 48 45 40 33 24 13

%e 14: 14 26 36 44 50 54 56 56 54 50 44 36 26 14

%e 15: 15 28 39 48 55 60 63 64 63 60 55 48 39 28 15

%e ... Formatted by _Wolfdieter Lang_, Dec 02 2014

%p seq(seq(i*(n-i),i=1..n-1),n=2..10); # _Robert Israel_, Dec 14 2015

%t Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* _Robert G. Wilson v_, Oct 06 2007 *)

%t 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 *)

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

%o (Magma) /* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // _Vincenzo Librandi_, Jul 12 2019

%Y Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.

%Y Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.

%Y Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

%K tabl,nonn,nice,easy,look

%O 1,2

%A _Marc LeBrun_

%E More terms from _Michael Somos_