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

1,2

Links

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

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)

Example

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;
  ...

Mathematica

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

Prog

(Magma) [n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // 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

Crossrefs

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

Cf. A128624 (row sums).

Sequence in context: A338449 A163890 A298983 * A182701 A277011 A277021

Adjacent sequences: A128620 A128621 A128622 * A128624 A128625 A128626

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Mar 14 2007

Extensions

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

Status

approved