A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

1, 2, 2, 6, 3, 3, 8, 8, 4, 4, 15, 10, 10, 5, 5, 18, 18, 12, 12, 6, 6, 28, 21, 21, 14, 14, 7, 7, 32, 32, 24, 24, 16, 16, 8, 8, 45, 36, 36, 27, 27, 18, 18, 9, 9, 50, 50, 40, 40, 30, 30, 20, 20, 10, 10, 66, 55, 55, 44, 44, 33, 33, 22, 22, 11, 11, 72, 72, 60, 60, 48, 48, 36, 36, 24, 24, 12, 12, 91, 78, 78, 65, 65, 52, 52, 39, 39, 26, 26, 13, 13

Offset: 1

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• easy
• tabl

Examples

			First few rows of the triangle are:
   1;
   2,  2;
   6,  3,  3;
   8,  8,  4,  4;
  15, 10, 10,  5,  5;
  18, 18, 12, 12,  6, 6;
  28, 21, 21, 14, 14, 7, 7;
  ...
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000027, A093005, A093353, A115514, A128621, A245524.

Cf. A128624 (row sums).

Programs

Magma

[n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Mathematica

Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

SageMath

flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Formula

Sum_{k=1..n} T(n, k) = A128624(n) (row sums).

T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - R. J. Mathar, Jun 27 2012

From G. C. Greubel, Mar 13 2024: (Start)

T(n, k) = n*A115514(n, k).

T(n, k) = Sum_{j=k..n} A128621(n, j).

T(n, 1) = A093005(n).

T(n, 2) = A093353(n-1), n >= 2.

T(n, n) = A000027(n).

T(2*n-1, n) = A245524(n).

Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

Extensions

a(41) = 27 inserted and more terms from Georg Fischer, Jun 05 2023