A123191 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).

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, -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, 1, -1, 3, -3, 3, -3, 3, -3, 3, -3, 3, -1

Offset: 0

Author

Roger L. Bagula, Oct 03 2006

Keywords

• sign
• tabl

Examples

			Triangle starts:
1;
-1,1;
-1,3,-1;
-1,3,-3,1;
-1,3,-3,3,-1;
-1,3,-3,3,-3,1;
		

Programs

Maple

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

Mathematica

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]

Formula

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)].

Extensions

Edited by N. J. A. Sloane, Oct 29 2006