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A128621 A127648 * A128174 as an infinite lower triangular matrix.

%I A128621 #18 Mar 22 2024 17:42:45
%S A128621 1,0,2,3,0,3,0,4,0,4,5,0,5,0,5,0,6,0,6,0,6,7,0,7,0,7,0,7,0,8,0,8,0,8,
%T A128621 0,8,9,0,9,0,9,0,9,0,9,0,10,0,10,0,10,0,10,0,10,11,0,11,0,11,0,11,0,
%U A128621 11,0,11,0,12,0,12,0,12,0,12,0,12,0,12,13,0,13,0,13,0,13,0,13,0,13,0,13
%N A128621 A127648 * A128174 as an infinite lower triangular matrix.
%H A128621 G. C. Greubel, <a href="/A128621/b128621.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A128621 Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...
%F A128621 T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.
%F A128621 Sum_{k=1..n} T(n, k) = A093005(n) (row sums).
%F A128621 From _G. C. Greubel_, Mar 13 2024: (Start)
%F A128621 T(n, k) = n*(1 + (-1)^(n+k))/2.
%F A128621 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).
%F A128621 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).
%F A128621 Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)
%e A128621 First few rows of the triangle:
%e A128621   1;
%e A128621   0, 2;
%e A128621   3, 0, 3;
%e A128621   0, 4, 0, 4;
%e A128621   5, 0, 5, 0, 5;
%e A128621   ...
%t A128621 Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 13 2024 *)
%o A128621 (Magma) [n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 13 2024
%o A128621 (SageMath) flatten([[n*(1+(-1)^(n+k))//2 for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 13 2024
%Y A128621 Cf. A000326, A123684, A127648, A128174.
%Y A128621 Cf. A093005 (row sums).
%K A128621 nonn,easy,tabl
%O A128621 1,3
%A A128621 _Gary W. Adamson_, Mar 14 2007
%E A128621 More terms added by _G. C. Greubel_, Mar 13 2024