This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380385 #20 Jan 25 2025 13:54:07
%S A380385 1,0,1,0,-1,1,0,0,-1,1,0,0,-1,-1,1,0,0,1,-1,-1,1,0,0,0,0,-1,-1,1,0,0,
%T A380385 0,1,0,-1,-1,1,0,0,0,1,0,0,-1,-1,1,0,0,0,-1,2,0,0,-1,-1,1,0,0,0,0,0,1,
%U A380385 0,0,-1,-1,1,0,0,0,0,0,1,1,0,0,-1,-1,1
%N A380385 Triangle read by rows. T(1, 1) = 1, T(n, k) = [n >= k](Sum_{i=1..k-1} T[n - i, k - 1] - Sum_{i=1..n-1} T[n - i, k]).
%C A380385 For n >= 1 the row sum of row n appears to be -A010815(n). See the Mathematica program for exact formulation of the recurrence.
%e A380385 {
%e A380385 {1},
%e A380385 {0, 1},
%e A380385 {0, -1, 1},
%e A380385 {0, 0, -1, 1},
%e A380385 {0, 0, -1, -1, 1},
%e A380385 {0, 0, 1, -1, -1, 1},
%e A380385 {0, 0, 0, 0, -1, -1, 1},
%e A380385 {0, 0, 0, 1, 0, -1, -1, 1},
%e A380385 {0, 0, 0, 1, 0, 0, -1, -1, 1},
%e A380385 {0, 0, 0, -1, 2, 0, 0, -1, -1, 1},
%e A380385 {0, 0, 0, 0, 0, 1, 0, 0, -1, -1, 1},
%e A380385 {0, 0, 0, 0, 0, 1, 1, 0, 0, -1, -1, 1}
%e A380385 }
%t A380385 nn = 12; t[n_, 1] = If[n == 1, 1, 0]; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, n - 1}], 0]; Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]; Flatten[%]
%t A380385 nn = 12; Table[Table[If[n == k, 1, Coefficient[Product[(1 - x^i), {i, k - 1}], x^(n - k)]], {k, 1, n}], {n, 1, nn}] // Flatten (* Conjectured after A231599. - _Mats Granvik_, Jan 25 2025 *)
%Y A380385 Cf. A231599, A010815, A008284.
%K A380385 sign,tabl
%O A380385 1,50
%A A380385 _Mats Granvik_, Jan 23 2025