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A139146 Interpolation one-half polynomials based on Chebyshev T(x.n) polynomial coefficients(A053120 ): even-> 2*T(x,n); odd->T(x,n) + T(x,n+1).

%I A139146 #5 Jan 16 2022 23:07:06
%S A139146 2,1,1,0,2,-1,1,2,-2,0,4,-1,-3,2,4,0,-6,0,8,1,-3,-8,4,8,2,0,-16,0,16,
%T A139146 1,5,-8,-20,8,16,0,10,0,-40,0,32
%N A139146 Interpolation one-half polynomials based on Chebyshev T(x.n) polynomial coefficients(A053120 ): even-> 2*T(x,n); odd->T(x,n) + T(x,n+1).
%C A139146 Row sums are all 2.
%C A139146 The rationale behind this interpolation is that Bessel functions have 1/2 values, so what about other orthogonal polynomials?
%C A139146 The integration shows that they are "mostly" orthogonal when three away from the diagonal.
%C A139146 TableForm[Table[Integrate[p[x, m]*p[x, n]/Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]]
%C A139146 These polynomials would also be related to two-dimensional Chladni-Chebyshev type standing waves as:
%C A139146 Chladni[x,y,n,m]=ChebyshevT[n, x] + ChebyshevT[m, y].
%F A139146 even->p(x.m)= 2*T(x,n); odd->p(x,m)=T(x,n)+T(x,n+1); out_n,m=Coefficients(p(x,m).
%e A139146 {2},
%e A139146 {1, 1},
%e A139146 {0, 2},
%e A139146 {-1, 1, 2},
%e A139146 {-2, 0, 4},
%e A139146 {-1, -3, 2, 4},
%e A139146 {0, -6, 0, 8},
%e A139146 {1, -3, -8, 4,8},
%e A139146 {2, 0, -16, 0, 16},
%e A139146 {1, 5, -8, -20, 8, 16},
%e A139146 {0, 10, 0, -40, 0, 32}
%t A139146 Clear[p, x] p[x, 0] = 2*ChebyshevT[0, x]; p[x, 1] = ChebyshevT[0, x] + ChebyshevT[1, x]; p[x, 2] = 2*ChebyshevT[1, x]; p[x_, m_] := p[x, m] = If[Mod[m, 2] == 0, 2*ChebyshevT[Floor[m/2], x], ChebyshevT[Floor[m/2], x] + ChebyshevT[Floor[m/2 + 1], x]]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]
%Y A139146 Cf. A053120.
%K A139146 uned,tabf,sign
%O A139146 1,1
%A A139146 _Roger L. Bagula_, Jun 05 2008