A246788 - OEIS

%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