%I #17 Oct 23 2014 20:50:09
%S 1,-1,1,-1,-7,1,-1,29,-17,1,-1,-99,175,-31,1,-1,301,-1425,569,-49,1,
%T -1,-851,10095,-8071,1391,-71,1,-1,2285,-65169,97769,-29969,2869,-97,
%U 1,-1,-5907,393583,-1063447,543471,-86731,5279,-127,1,-1,14829,-2260625,10693865,-8746257,2181269,-212449,8945,-161,1,-1,-36371,12484975,-101280535,128879343,-48218731,7045151,-461455,14239,-199,1
%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+2k)^k for 0 <= k <= n .
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+2)^1 + A_2*(x+4)^2 + ... + A_n*(x+2n)^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 - 2*n^2 for n > 0.
%e 1;
%e -1, 1;
%e -1, -7, 1;
%e -1, 29, -17, 1;
%e -1, -99, 175, -31, 1;
%e -1, 301, -1425, 569, -49, 1;
%e -1, -851, 10095, -8071, 1391, -71, 1;
%e -1, 2285, -65169, 97769, -29969, 2869, -97, 1;
%e -1, -5907, 393583, -1063447, 543471, -86731, 5279, -127, 1;
%e -1, 14829, -2260625, 10693865, -8746257, 2181269, -212449, 8945, -161, 1;
%o (PARI) for(n=0,20,for(k=0,n,if(!k,if(n,print1(-1,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,((-2*k)^(i-k)*i*binomial(i,k)))/k,", "))))
%Y Cf. A056220, A248826, A248830.
%K sign,tabl
%O 0,5
%A _Derek Orr_, Oct 15 2014