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A248975 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.

%I A248975 #9 Oct 23 2014 20:50:41
%S A248975 1,2,1,-1,-1,1,-12,-10,4,1,45,34,-14,-3,1,406,319,-124,-33,6,1,-2357,
%T A248975 -1847,731,187,-39,-5,1,-26968,-21188,8312,2182,-424,-68,8,1,223769,
%U A248975 175700,-69052,-18034,3566,548,-76,-7,1,3096810,2432333,-955048,-250126,49052,7730,-1000,-115,10,1
%N A248975 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 A248975 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 A248975 T(n,n-1) = 1 - n*(-1)^n for n > 0.
%F A248975 T(n,n-2) = (1-n)*((3/2)*n-(-1)^n) + 1 for n > 1.
%F A248975 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 A248975 1;
%e A248975 2,             1;
%e A248975 -1,           -1,       1;
%e A248975 -12,         -10,       4,       1;
%e A248975 45,           34,     -14,      -3,     1;
%e A248975 406,         319,    -124,     -33,     6,    1;
%e A248975 -2357,     -1847,     731,     187,   -39,   -5,     1;
%e A248975 -26968,   -21188,    8312,    2182,  -424,  -68,     8,    1;
%e A248975 223769,   175700,  -69052,  -18034,  3566,  548,   -76,   -7,  1;
%e A248975 3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1
%o A248975 (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 A248975 for(n=0,15,for(j=0,n,print1(a(n,j),", ")))
%K A248975 sign,tabl
%O A248975 0,2
%A A248975 _Derek Orr_, Oct 18 2014