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

%I A137422 #17 Apr 02 2022 17:46:50
%S A137422 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,
%T A137422 -1,-1,18,18,-48,-48,32,32,0,-7,-7,56,56,-112,-112,64,64,1,1,-32,-32,
%U A137422 160,160,-256,-256,128,128,0,9,9,-120,-120,432,432,-576,-576,256,256
%N A137422 Triangle T(n,k) = A053120(n-1,k) + A053120(n-1,k-1), read by rows.
%C A137422 Row sums are 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e A137422 Triangle begins
%e A137422    0;
%e A137422    1,  1;
%e A137422    0,  1,   1;
%e A137422   -1, -1,   2,    2;
%e A137422    0, -3,  -3,    4,    4;
%e A137422    1,  1,  -8,   -8,    8,    8;
%e A137422    0,  5,   5,  -20,  -20,   16,   16;
%e A137422   -1, -1,  18,   18,  -48,  -48,   32,   32;
%e A137422    0, -7,  -7,   56,   56, -112, -112,   64,   64;
%e A137422    1,  1, -32,  -32,  160,  160, -256, -256,  128, 128;
%e A137422    0,  9,   9, -120, -120,  432,  432, -576, -576, 256, 256;
%p A137422 A053120 := proc(n, k)
%p A137422     if n <0 or k <0 then
%p A137422         0 ;
%p A137422     else
%p A137422         T(n, x) ;
%p A137422         coeftayl(%, x=0, k) ;
%p A137422     end if;
%p A137422 end proc:
%p A137422 A137422 := proc(n,k)
%p A137422     A053120(n-1,k)+A053120(n-1,k-1)
%p A137422 end proc: # _R. J. Mathar_, Sep 10 2013
%t A137422 (* 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]
%Y A137422 Cf. A053120.
%K A137422 tabl,sign
%O A137422 0,9
%A A137422 _Roger L. Bagula_ and _Gary W. Adamson_, Apr 16 2008
%E A137422 T(0,0) set to a rationalized 0. - _R. J. Mathar_, Sep 10 2013