%I #9 Oct 23 2014 20:50:41
%S 1,2,1,-1,-1,1,-12,-10,4,1,45,34,-14,-3,1,406,319,-124,-33,6,1,-2357,
%T -1847,731,187,-39,-5,1,-26968,-21188,8312,2182,-424,-68,8,1,223769,
%U 175700,-69052,-18034,3566,548,-76,-7,1,3096810,2432333,-955048,-250126,49052,7730,-1000,-115,10,1
%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 T(n,0) = 1 - sum(i=1..n) (-1)^i*T(n,i) = 1 + T(n,1) - T(n-2) + T(n-3) - ... + (-1)^(n-1)*T(n,n-1) + (-1)^n.
%e 1;
%e 2, 1;
%e -1, -1, 1;
%e -12, -10, 4, 1;
%e 45, 34, -14, -3, 1;
%e 406, 319, -124, -33, 6, 1;
%e -2357, -1847, 731, 187, -39, -5, 1;
%e -26968, -21188, 8312, 2182, -424, -68, 8, 1;
%e 223769, 175700, -69052, -18034, 3566, 548, -76, -7, 1;
%e 3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1
%o (PARI) a(n,j) = if(j==n,return(1));if(j!=n,return(1-sum(i=1,n-j,(-1)^(i*(j+1))*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 0,2
%A _Derek Orr_, Oct 18 2014