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Product of number triangles A127243 and A127248.
1

%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