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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)=((-1)^(n-1)-x)*p(n-1,x)-p(n-2,x) for n>=2 (0<=k<=n).
1

%I #15 Sep 17 2024 08:25:10

%S 1,-1,-1,0,2,1,1,3,-1,-1,-1,-6,-3,2,1,-2,-8,4,6,-1,-1,3,16,7,-12,-6,2,

%T 1,5,21,-13,-25,7,9,-1,-1,-8,-42,-15,50,24,-18,-9,2,1,-13,-55,40,90,

%U -33,-51,10,12,-1,-1,21,110,30,-180,-81,102,50,-24,-12,2,1

%N 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)=((-1)^(n-1)-x)*p(n-1,x)-p(n-2,x) for n>=2 (0<=k<=n).

%H Joanne Dombrowski, <a href="http://projecteuclid.org/euclid.pjm/1102703882">Tridiagonal matrix representations of cyclic selfadjoint operators</a>, Pacific J. Math. 114, no. 2 (1984), 325-334.

%e Triangle begins:

%e {1},

%e {-1, -1},

%e {0, 2, 1},

%e {1, 3, -1, -1},

%e {-1, -6, -3, 2, 1},

%e {-2, -8, 4, 6, -1, -1},

%e {3, 16, 7, -12, -6, 2, 1},

%e {5, 21, -13, -25, 7, 9, -1, -1},

%e {-8, -42, -15, 50, 24, -18, -9, 2, 1},

%e {-13, -55, 40, 90, -33, -51, 10, 12, -1, -1},

%e {21, 110, 30, -180, -81, 102, 50, -24, -12, 2, 1}

%p p[0]:=1: p[1]:=-1-x: for n from 2 to 12 do p[n]:=sort(expand(((-1)^(n-1)-x)*p[n-1]-p[n-2])) od: T:=(n,k)->coeff(p[n],x,k): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

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

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 01 2006

%E Edited by _N. J. A. Sloane_, Dec 02 2006