A246798 - OEIS

%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