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%I #14 Nov 21 2014 02:21:28
%S 1,-3,2,9,-10,3,-23,38,-21,4,57,-122,99,-36,5,-135,358,-381,204,-55,6,
%T 313,-986,1299,-916,365,-78,7,-711,2598,-4077,3564,-1875,594,-105,8,
%U 1593,-6618,12051,-12564,8205,-3438,903,-136,9,-3527,16422,-34029,41196,-32115,16722,-5817,1304,-171,10
%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+2)^k.
%C Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+2)^0 + A_1*(x+2)^1 + A_2*(x+2)^2 + ... + A_n*(x+2)^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,0) = ((6*n+8)*(-2)^n+1)/9, for n >= 0.
%F T(n,n-1) = -n*(2*n+1), for n >= 1.
%F Row n sums to A001057(n+1).
%e 1;
%e -3, 2;
%e 9, -10, 3;
%e -23, 38, -21, 4;
%e 57, -122, 99, -36, 5;
%e -135, 358, -381, 204, -55, 6;
%e 313, -986, 1299, -916, 365, -78, 7;
%e -711, 2598, -4077, 3564, -1875, 594, -105, 8;
%e 1593, -6618, 12051, -12564, 8205, -3438, 903, -136, 9;
%e -3527, 16422, -34029, 41196, -32115, 16722, -5817, 1304, -171, 10;
%o (PARI) T(n,k) = (k+1)*sum(i=0,n-k,(-2)^i*binomial(i+k+1,k+1))
%o for(n=0,10,for(k=0,n,print1(T(n,k),", ")))
%Y Cf. A248345, A045883, A014105, A001057.
%K sign,tabl
%O 0,2
%A _Derek Orr_, Nov 15 2014
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