A137422 Triangle T(n,k) = A053120(n-1,k) + A053120(n-1,k-1), read by rows.

0, 1, 1, 0, 1, 1, -1, -1, 2, 2, 0, -3, -3, 4, 4, 1, 1, -8, -8, 8, 8, 0, 5, 5, -20, -20, 16, 16, -1, -1, 18, 18, -48, -48, 32, 32, 0, -7, -7, 56, 56, -112, -112, 64, 64, 1, 1, -32, -32, 160, 160, -256, -256, 128, 128, 0, 9, 9, -120, -120, 432, 432, -576, -576, 256, 256

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Apr 16 2008

Keywords

• tabl
• sign

Comments

Row sums are 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...

Examples

			Triangle begins
   0;
   1,  1;
   0,  1,   1;
  -1, -1,   2,    2;
   0, -3,  -3,    4,    4;
   1,  1,  -8,   -8,    8,    8;
   0,  5,   5,  -20,  -20,   16,   16;
  -1, -1,  18,   18,  -48,  -48,   32,   32;
   0, -7,  -7,   56,   56, -112, -112,   64,   64;
   1,  1, -32,  -32,  160,  160, -256, -256,  128, 128;
   0,  9,   9, -120, -120,  432,  432, -576, -576, 256, 256;
		

Crossrefs

Cf. A053120.

Programs

Maple

A053120 := proc(n, k)
    if n <0 or k <0 then
        0 ;
    else
        T(n, x) ;
        coeftayl(%, x=0, k) ;
    end if;
end proc:
A137422 := proc(n,k)
    A053120(n-1,k)+A053120(n-1,k-1)
end proc: # R. J. Mathar, Sep 10 2013

Mathematica

(* Chebyshev A053120 polynomials*) (* Recursive root shifted polynomials*) Q[x, 0] = 1; Q[x, 1] = x + 1; Q[x_, n_] := (x + 1)*ChebyshevT[n - 1, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

Extensions

T(0,0) set to a rationalized 0. - R. J. Mathar, Sep 10 2013