A104633 Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.

1, 3, 2, 6, 6, 3, 10, 12, 9, 4, 15, 20, 18, 12, 5, 21, 30, 30, 24, 15, 6, 28, 42, 45, 40, 30, 18, 7, 36, 56, 63, 60, 50, 36, 21, 8, 45, 72, 84, 84, 75, 60, 42, 24, 9, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 66, 110, 135

Offset: 1

Author

Gary W. Adamson, Mar 18 2005

Keywords

• nonn
• tabl
• easy

Comments

The triangle can be constructed multiplying the triangle A(n,k)=n-k+1 (if 1<=k<=n, else 0) by the triangle B(n,k) =k (if 1<=k<=n, else 0).

Swapping the two triangles of this matrix product would generate A104634.

Examples

			First few rows of the triangle:
  1;
  3,  2;
  6,  6,  3;
 10, 12,  9,  4;
 15, 20, 18, 12,  5;
 21, 30, 30, 24, 15,  6;
 28, 42, 45, 40, 30, 18,  7;
 36, 56, 63, 60, 50, 36, 21, 8;
 ...
e.g. Col. 3 = 3 * (1, 3, 6, 10, 15...) = 3, 9, 18, 30, 45...
		

Links

• G. C. Greubel, Rows n=1..100 of triangle, flattened
• Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
• Joaquín Figueroa, Ivan Gonzalez, and Daniel Salinas-Arizmendi, A Novel Transfer Matrix Framework for Multiple Dirac Delta Potentials, arXiv:2503.23134 [quant-ph], 2025. See pp. 4, 9.

Crossrefs

Cf. A062707, A158824, A104634, A001296, A000332 (row sums).

Programs

Magma

[[k*(k-n-1)*(k-n-2)/2: k in [1..n]]: n in [1..20]]; // G. C. Greubel, Aug 12 2018

Maple

A104633 := proc(n,k) k*(k-n-1)*(k-n-2)/2 ; end proc:
seq(seq(A104633(n,k),k=1..n),n=1..16) ; # R. J. Mathar, Mar 03 2011

Mathematica

Table[k*(k-n-1)*(k-n-2)/2, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Aug 12 2018 *)

PARI

for(n=1,20, for(k=1,n, print1(k*(k-n-1)*(k-n-2)/2, ", "))) \\ G. C. Greubel, Aug 12 2018

Formula

G.f.: x*y/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, May 22 2023