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A143218
Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.
2
1, 3, 9, 5, 15, 25, 7, 21, 35, 49, 9, 27, 45, 63, 81, 11, 33, 55, 77, 99, 121, 13, 39, 65, 91, 117, 143, 169, 15, 45, 75, 105, 135, 165, 195, 225, 17, 51, 85, 119, 153, 187, 221, 255, 289, 19, 57, 95, 133, 171, 209, 247, 285, 323, 361, 21, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441
OFFSET
1,2
FORMULA
Triangle read by rows, A127775 * A000012 * A127775.
T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.
Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A131507(n,k) * A127775(n,k).
T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.
T(2*n-1, n) = A014634(n-1), n >= 1.
T(2*n-2, n-1) = A033567(n-1), n >= 2.
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)
EXAMPLE
First few rows of the triangle =
1;
3, 9;
5, 15, 25;
7, 21, 35, 49;
9, 27, 45, 63, 81;
11, 33, 55, 77, 99, 121;
13, 39, 65, 91, 117, 143, 169;
...
T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).
MATHEMATICA
Table[(2*k-1)*(2*n-1), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)
PROG
(Magma) [(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022
(SageMath) flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jul 30 2008
STATUS
approved