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%I #13 Mar 22 2024 17:41:51
%S 1,0,1,1,0,2,0,1,0,3,1,0,2,0,5,0,1,0,3,0,8,1,0,2,0,5,0,13,0,1,0,3,0,8,
%T 0,21,1,0,2,0,5,0,13,0,34,0,1,0,3,0,8,0,21,0,55,1,0,2,0,5,0,13,0,34,0,
%U 89,0,1,0,3,0,8,0,21,0,55,0,144,1,0,2,0,5,0,13,0,34,0,89,0,233
%N Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.
%C This triangle is different from A128619, which is A128619 = A127647 * A128174.
%H G. C. Greubel, <a href="/A128618/b128618.txt">Rows n = 1..100 of the triangle, flattened</a>
%F By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
%F Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
%F From _G. C. Greubel_, Mar 17 2024: (Start)
%F T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
%F T(n, n) = A000045(n).
%F T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
%F Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
%F Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) * A355020(floor((n-1)/2)). (End)
%e First few rows of the triangle are:
%e 1;
%e 0, 1;
%e 1, 0, 2;
%e 0, 1, 0, 3;
%e 1, 0, 2, 0, 5;
%e 0, 1, 0, 3, 0, 8;
%e 1, 0, 2, 0, 5, 0, 13;
%e 0, 1, 0, 3, 0, 8, 0, 21;
%e 1, 0, 2, 0, 5, 0, 13, 0, 34;
%e 0, 1, 0, 3, 0, 8, 0, 21, 0, 55;
%e 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89;
%e ...
%t Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 17 2024 *)
%o (Magma) [((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 17 2024
%o (SageMath) flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 17 2024
%Y Cf. A000045, A052952, A127647, A128174, A128619, A355020.
%K nonn,tabl
%O 1,6
%A _Gary W. Adamson_, Mar 14 2007
%E a(6) corrected and more terms from _Georg Fischer_, May 30 2023
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