A249266 - OEIS

%I #11 Nov 20 2014 19:36:30

%S 1,3,1,-9,-3,1,-97,-39,7,1,815,313,-65,-7,1,12367,4873,-945,-127,11,1,

%T -164465,-64439,12735,1633,-169,-11,1,-3314673,-1302263,255327,33553,

%U -3249,-263,15,1,60873999,23899401,-4695969,-613359,60591,4665,-321,-15,1

%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k*(x+2*(-1)^k)^k.

%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+2)^0 + A_1*(x-2)^1 + A_2*(x+2)^2 + A_3*(x-2)^3 + ... + A_n*(x+2*(-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 T(n,n-1) = 1-2*n*(-1)^n, for n > 0.

%e 1;

%e 3,               1;

%e -9,             -3,        1;

%e -97,           -39,        7,       1;

%e 815,           313,      -65,      -7,     1;

%e 12367,        4873,     -945,    -127,    11,    1;

%e -164465,    -64439,    12735,    1633,  -169,  -11,    1;

%e -3314673, -1302263,   255327,   33553, -3249, -263,   15,   1;

%e 60873999, 23899401, -4695969, -613359, 60591, 4665, -321, -15, 1;

%o (PARI) a(n,j,L)=if(j==n,return(1));if(j!=n,return(1-sum(i=1,n-j,(-L)^i*(-1)^(i*j)*binomial(i+j,i)*a(n,i+j,L))))

%o for(n=0,10,for(j=0,n,print1(a(n,j,2),", ")))

%Y Cf. A248975.

%K sign,tabl

%O 0,2

%A _Derek Orr_, Oct 23 2014