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A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

%I A143219 #12 Jul 13 2022 03:41:45
%S A143219 1,2,6,3,9,18,4,12,24,40,5,15,30,50,75,6,18,36,60,90,126,7,21,42,70,
%T A143219 105,147,196,8,24,48,80,120,168,224,288,9,27,54,90,135,189,252,324,
%U A143219 405,10,30,60,100,150,210,280,360,450,550
%N A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.
%H A143219 G. C. Greubel, <a href="/A143219/b143219.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.
%F A143219 Sum_{k=1..n} T(n, k) = A002417(n).
%F A143219 T(n, n) = A002411(n).
%F A143219 From _G. C. Greubel_, Jul 12 2022: (Start)
%F A143219 T(n, k) = A002024(n,k) * A127773(n,k).
%F A143219 T(n, k) = n * binomial(k+1, 2).
%F A143219 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/4)*(4*n - 3*floor((n+1)/2) + 3)*binomial(2 + floor((n+1)/2), 3).
%F A143219 T(2*n-1, n) = A002414(n), n >= 1.
%F A143219 T(2*n-2, n-1) = A011379(n-1), n >= 2. (End)
%e A143219 First few rows of the triangle =
%e A143219   1;
%e A143219   2,  6;
%e A143219   3,  9, 18;
%e A143219   4, 12, 24, 40;
%e A143219   5, 15, 30, 50,  75;
%e A143219   6, 18, 36, 60,  90, 126;
%e A143219   7, 21, 42, 70, 105, 147, 196;
%e A143219   ...
%t A143219 Table[n*Binomial[k+1, 2], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 12 2022 *)
%o A143219 (Magma) [n*Binomial(k+1, 2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 12 2022
%o A143219 (SageMath) flatten([[n*binomial(k+1, 2) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jul 12 2022
%Y A143219 Cf. A000012, A127648, A127773.
%Y A143219 Cf. A002024, A002411 (right border), A002414, A002417 (row sums), A011379.
%K A143219 nonn,tabl
%O A143219 1,2
%A A143219 _Gary W. Adamson_, Jul 30 2008