%I #9 Mar 27 2022 23:29:59
%S -1,-1,2,1,2,-3,-1,-2,-3,4,1,2,3,4,-5,-1,-2,-3,-4,-5,6,1,2,3,4,5,6,-7,
%T -1,-2,-3,-4,-5,-6,-7,8,1,2,3,4,5,6,7,8,-9,-1,-2,-3,-4,-5,-6,-7,-8,-9,
%U 10,1,2,3,4,5,6,7,8,9,10,-11
%N Triangle T(n,k) = k*A122188(n,k), read by rows.
%C Multiplication of the columns of A122188 by their index is equivalent to differentiation of the polynomials B(n,x) defined in A122188.
%C Row sums are -1, 1, 0, -2, 5, -9, 14, -20, 27, -35, 44, ... =(-1)^n*A080956(n-1).
%F |T(n,k)| = A002260(n,k).
%e -1;
%e -1, 2;
%e 1, 2, -3;
%e -1, -2, -3, 4;
%e 1, 2, 3, 4, -5;
%e -1, -2, -3, -4, -5, 6;
%e 1, 2, 3, 4, 5, 6, -7;
%e -1, -2, -3, -4, -5, -6, -7, 8;
%e 1, 2, 3, 4, 5, 6, 7, 8, -9;
%e -1, -2, -3, -4, -5, -6, -7, -8, -9, 10;
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -11;
%t Clear[B, x, n] B[x, 0] = 1; B[x, 1] = -x + 1; B[x_, n_] := B[x, n] = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]; P[x_, n_] := D[B[x, n + 1], x]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];
%Y Cf. A002260, A122188.
%K easy,tabl,sign
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 18 2008