A140056 - OEIS

%I #12 Feb 17 2024 23:44:03

%S -3,-2,-1,-1,-2,-1,-1,-1,-1,-1,0,-1,-1,-1,0,0,-1,-1,-1,0,0,0,-1,-1,-1,

%T 0,0,0,0,-1,-1,-1,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,

%U 0,-1

%N Triangle of coefficients: f(x,y,n) = x^n - y^(n-1)*x - y^n; p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n).

%C Row sums are all -3.

%H Eric Weisstein's World of Mathematics <a href="http://mathworld.wolfram.com/KleinQuartic.html">Klein Quartic</a>.

%F f(x,y,n)=x^n - y^(n - 1)*x - y^n; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n);

%e {-3},

%e {-2, -1},

%e {-1, -2},

%e {-1, -1, -1},

%e {-1, -1, 0, -1},

%e {-1, -1, 0, 0, -1},

%e {-1, -1, 0, 0, 0, -1},

%e {-1, -1, 0, 0, 0, 0, -1},

%e {-1, -1, 0, 0, 0, 0, 0, -1},

%e {-1, -1, 0, 0, 0, 0, 0, 0, -1},

%e {-1, -1, 0, 0, 0, 0, 0, 0, 0, -1}.

%e Polynomials before lower to x only are:

%e -3,

%e -x - y - z,

%e -xy - x z - y z,

%e -xy^2 - x^2 z - y z^2,

%e -x y^3 - x^3 z - y z^3,

%e -x y^4 - x^4 z - y z^4,

%e ...

%t f[x_, y_, n] = If[n > 0, x^n - y^(n - 1)*x - y^n, -1]; p[x_, y_, z_, n_] = f[x, y, n] + f[y, z, n] + f[z, x, n];

%t Table[ExpandAll[p[x, y, z, n]], {n, 0, 10}];

%t a = Table[CoefficientList[p[x, y, z, n] /. y -> 1 /. z -> 1, x], {n, 0, 10}];

%t Flatten[a]

%t Table[Apply[Plus, CoefficientList[p[x, y, z, n] /. y -> 1 /. z -> 1, x]], {n, 0, 10}];

%K uned,tabl,sign

%O 1,1

%A _Roger L. Bagula_, Jun 14 2008