A128621 - OEIS

%I #18 Mar 22 2024 17:42:45

%S 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 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 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 A127648 * A128174 as an infinite lower triangular matrix.

%H G. C. Greubel, <a href="/A128621/b128621.txt">Rows n = 1..100 of the triangle, flattened</a>

%F Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...

%F T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.

%F Sum_{k=1..n} T(n, k) = A093005(n) (row sums).

%F From _G. C. Greubel_, Mar 13 2024: (Start)

%F T(n, k) = n*(1 + (-1)^(n+k))/2.

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).

%F Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).

%F 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 First few rows of the triangle:

%e   1;

%e   0, 2;

%e   3, 0, 3;

%e   0, 4, 0, 4;

%e   5, 0, 5, 0, 5;

%e   ...

%t Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 13 2024 *)

%o (Magma) [n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 13 2024

%o (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 Cf. A000326, A123684, A127648, A128174.

%Y Cf. A093005 (row sums).

%K nonn,easy,tabl

%O 1,3

%A _Gary W. Adamson_, Mar 14 2007

%E More terms added by _G. C. Greubel_, Mar 13 2024