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%I #6 Oct 23 2014 20:51:06
%S 1,0,1,-3,3,1,8,-6,-2,1,65,-50,-20,5,1,-296,235,90,-25,-4,1,-3059,
%T 2401,945,-245,-49,7,1,21552,-16940,-6636,1750,336,-56,-6,1,272289,
%U -213828,-84000,21966,4326,-672,-90,9,1,-2600752,2042805,801996,-210126,-41160,6570,834,-99,-8,1
%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-(-1)^k)^k for 0 <= k <= n.
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x+1)^1 + A_2*(x-1)^2 + A_3*(x+1)^3 + ... + A_n*(x-(-1)^n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
%F T(n,n-1) = 1 + n*(-1)^n for n > 0.
%F T(n,n-2) = (1-n)*((3/2)*n+(-1)^n) + 1 for n > 1.
%F Rows sum to 1.
%e 1;
%e 0, 1;
%e -3, 3, 1;
%e 8, -6, -2, 1;
%e 65, -50, -20, 5, 1;
%e -296, 235, 90, -25, -4, 1;
%e -3059, 2401, 945, -245, -49, 7, 1;
%e 21552, -16940, -6636, 1750, 336, -56, -6, 1;
%e 272289, -213828, -84000, 21966, 4326, -672, -90, 9, 1;
%e -2600752, 2042805, 801996, -210126, -41160, 6570, 834, -99, -8, 1
%o (PARI) a(n,j)=if(j==n,return(1));if(j!=n,return(1-sum(i=1,n-j,(-1)^(i*j)*binomial(i+j,i)*a(n,i+j))))
%o for(n=0, 15, for(j=0, n, print1(a(n, j), ", ")))
%K sign,tabl
%O 0,4
%A _Derek Orr_, Oct 18 2014
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