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%I #13 Nov 20 2014 19:37:03
%S 1,-1,1,-13,5,1,75,-31,-5,1,987,-383,-77,9,1,-10565,4177,803,-111,-9,
%T 1,-187397,73489,14483,-1871,-189,13,1,2962811,-1164335,-228109,30049,
%U 2891,-239,-13,1,67151483,-26365999,-5179405,676961,66731,-5167,-349,17,1
%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k*(x-2*(-1)^k)^k.
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-2)^0 + A_1*(x+2)^1 + A_2*(x-2)^2 + A_3*(x+2)^3 + ... + A_n*(x-2*(-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+2*n*(-1)^n, for n > 0.
%e 1;
%e -1, 1;
%e -13, 5, 1;
%e 75, -31, -5, 1;
%e 987, -383, -77, 9, 1;
%e -10565, 4177, 803, -111, -9, 1;
%e -187397, 73489, 14483, -1871, -189, 13, 1;
%e 2962811, -1164335, -228109, 30049, 2891, -239, -13, 1;
%e 67151483, -26365999, -5179405, 676961, 66731, -5167, -349, 17, 1;
%o (PARI) a(n, j, L)=if(j==n, return(1)); if(j!=n, return(1-sum(i=1, n-j, (-L)^i*(-1)^(i*j)*binomial(i+j, i)*a(n, i+j, L))))
%o for(n=0, 10, for(j=0, n, print1(a(n, j, -2), ", ")))
%Y Cf. A248976.
%K sign,tabl
%O 0,4
%A _Derek Orr_, Oct 23 2014