%I #7 Aug 04 2023 04:35:43
%S 1,0,1,-1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T 0,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1
%N Product of number triangles A127243 and A127248.
%C Rows containing -1 entries are indexed by twice the odious numbers given by A091855.
%e Triangle begins:
%e 1;
%e 0, 1;
%e -1, 0, 1;
%e 0, 0, 0, 1;
%e 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, -1, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e ...
%t T1[n_, k_] := SeriesCoefficient[(1 + ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127243 *)
%t T2[n_, k_] := SeriesCoefficient[(1 - ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127248 *)
%t T[n_, k_] := Sum[T2[n, j]*T1[j, k], {j, 0, n}];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 04 2023 *)
%Y Row sums are A127254.
%Y Cf. A000069, A091855, A127243, A127248.
%K sign,tabl
%O 0,1
%A _Paul Barry_, Jan 10 2007