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%I #16 Nov 22 2014 03:50:29
%S 1,-5,2,22,-16,3,-86,92,-33,4,319,-448,237,-56,5,-1139,1982,-1383,484,
%T -85,6,3964,-8224,7122,-3296,860,-120,7,-13532,32600,-33702,19384,
%U -6700,1392,-161,8,45517,-124864,150006,-103088,44330,-12216,2107,-208,9,-151313,465626,-637314,509272,-261850,89844,-20573,3032,-261,10
%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+3)^k.
%C Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^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) = (1-(4*n+5)*(-3)^(n+1))/16, for n >= 0.
%F T(n,n-1) = -n*(3*n+2), for n >= 1.
%F Row n sums to (-1)^n*A045883(n+1) = T(n,0) of A246788.
%e Triangle starts:
%e 1;
%e -5, 2;
%e 22, -16, 3;
%e -86, 92, -33, 4;
%e 319, -448, 237, -56, 5;
%e -1139, 1982, -1383, 484, -85, 6;
%e 3964, -8224, 7122, -3296, 860, -120, 7;
%e -13532, 32600, -33702, 19384, -6700, 1392, -161, 8;
%e 45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9;
%e -151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 10;
%e ...
%o (PARI) T(n, k) = (k+1)*sum(i=0, n-k, (-3)^i*binomial(i+k+1, k+1))
%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
%Y Cf. A246788, A045944, A191008.
%K sign,tabl
%O 0,2
%A _Derek Orr_, Nov 15 2014
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