A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0

Offset: 0

Author

Paul Curtz, Jun 24 2011

Keywords

• sign
• tabl
• easy

Comments

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):

0, -1, -1, -1, -1, -1, -1, -1,

1, 0, -1, -2, -3, -4, -5, -6,

1, 1, 0, -2, -5, -9, -14, -20,

1, 2, 2, 0, -5, -14, -28, -48,

1, 3, 5, 5, 0, -14, -42, -90,

1, 4, 9, 14, 14, 0, -42, -132,

1, 5, 14, 28, 42, 42, 0, -132,

1, 6, 20, 48, 90, 132, 132, 0.

The rows in this array are essentially the columns of T(n,k).

Examples

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;
		

Crossrefs

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

Programs

Maple

p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Formula

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.