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A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

%I A143218 #5 Jul 12 2022 15:30:04
%S A143218 1,3,9,5,15,25,7,21,35,49,9,27,45,63,81,11,33,55,77,99,121,13,39,65,
%T A143218 91,117,143,169,15,45,75,105,135,165,195,225,17,51,85,119,153,187,221,
%U A143218 255,289,19,57,95,133,171,209,247,285,323,361,21,63,105,147,189,231,273,315,357,399,441
%N A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.
%H A143218 G. C. Greubel, <a href="/A143218/b143218.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A143218 Triangle read by rows, A127775 * A000012 * A127775.
%F A143218 T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.
%F A143218 Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).
%F A143218 From _G. C. Greubel_, Jul 12 2022: (Start)
%F A143218 T(n, k) = A131507(n,k) * A127775(n,k).
%F A143218 T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.
%F A143218 T(2*n-1, n) = A014634(n-1), n >= 1.
%F A143218 T(2*n-2, n-1) = A033567(n-1), n >= 2.
%F A143218 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)
%e A143218 First few rows of the triangle =
%e A143218    1;
%e A143218    3,  9;
%e A143218    5, 15, 25;
%e A143218    7, 21, 35, 49;
%e A143218    9, 27, 45, 63,  81;
%e A143218   11, 33, 55, 77,  99, 121;
%e A143218   13, 39, 65, 91, 117, 143, 169;
%e A143218   ...
%e A143218 T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).
%t A143218 Table[(2*k-1)*(2*n-1), {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 12 2022 *)
%o A143218 (Magma) [(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 12 2022
%o A143218 (SageMath) flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jul 12 2022
%Y A143218 Cf. A000012, A005408, A014634, A015237 (row sums).
%Y A143218 Cf. A016754, A024598, A033567, A127775, A131507.
%K A143218 nonn,tabl
%O A143218 1,2
%A A143218 _Gary W. Adamson_, Jul 30 2008