A123230 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x)=1, p(1,x)=1+x, p(n,x)=xp(n-1,x)+(-1)^(n+1)p(n-2,x) for n>=2.
1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 0, -1, 1, 1, 1, 1, 1, 0, 1, 1, -1, 1, 2, 0, -1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, -2, 2, 3, 0, -1, 1, 1, 1, 1, 2, 0, 3, 3, 1, 0, 1, 1, -1, 1, 3, 0, -3, 3, 4, 0, -1, 1, 1
Offset: 0
Examples
{1},
{1, 1},
{-1, 1, 1},
{1, 0, 1, 1},
{1, 0, -1, 1, 1},
{1, 1, 1, 0, 1, 1},
{-1, 1, 2, 0, -1, 1, 1},
{1, 0, 2, 2, 1, 0, 1, 1},
{1, 0, -2, 2, 3, 0, -1, 1,1},
{1, 1, 2, 0, 3, 3, 1, 0, 1, 1},
{-1, 1, 3, 0, -3, 3, 4, 0, -1, 1, 1}
Programs
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Maple
P[0]:=1: P[1]:=1+x: for n from 2 to 14 do P[n]:=sort(expand(x*P[n-1]+(-1)^(n+1)*P[n-2])) od: for n from 0 to 14 do seq(coeff(P[n],x,k),k=0..n) od; # yields sequence in triangular form
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Mathematica
p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = x*p[k - 1, x] + (-1)^(n + 1)p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
Extensions
Edited by N. J. A. Sloane, Oct 29 2006