A123191 - OEIS

%I #7 Mar 12 2014 16:37:00

%S 1,-1,1,-1,3,-1,-1,3,-3,1,-1,3,-3,3,-1,-1,3,-3,3,-3,1,-1,3,-3,3,-3,3,

%T -1,-1,3,-3,3,-3,3,-3,1,-1,3,-3,3,-3,3,-3,3,-1,-1,3,-3,3,-3,3,-3,3,-3,

%U 1,-1,3,-3,3,-3,3,-3,3,-3,3,-1

%N Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P[n] defined by P[0]=1, P[1]=x-1, P[n]=(1-x)P[n-1]+xP[n-2] for n>=2. Alternatively, P[n]=-1-(-x)^n-3*Sum((-x)^k,k=1..n-1).

%F T(0,0)=1; T(n,n)=(-1)^(n+1) for n>=1; T(n,0)=-1 for n>=1; T(n,k)=(-1)^(k+1)*3 for n>=2, 1<=k<=n-1. G.f.=G(t,x)=(1+2tx-2x)/[(1-x)(1+tx)].

%e Triangle starts:

%e 1;

%e -1,1;

%e -1,3,-1;

%e -1,3,-3,1;

%e -1,3,-3,3,-1;

%e -1,3,-3,3,-3,1;

%p T:=proc(n,k): if n=0 and k=0 then 1 elif k=n then (-1)^(n+1) elif k=0 then -1 else (-1)^(k+1)*3 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%t p[0, x] = 1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = (1 - x)*p[k - 1, x] + x*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Oct 03 2006

%E Edited by _N. J. A. Sloane_, Oct 29 2006