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A137422
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A triangular sequence based on concepts of operations on existing sequences: in this case the T(x,n) ( A053120 ) traditional Chebyshev of the first kind are "shifted" to give a new sequence: q(x,n)=(x-1)*T(x,n-1).
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1, 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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,9
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COMMENTS
| Row sums are:
{1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}.
These polynomials are based on the ideas of Gary Adamson
and concepts of "adjusted" sequences.
In this specific case the root structure is shifted up one and and "edge" root is added.
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FORMULA
| q(x,0}=1;q(x,1)=x+1; q(x,n)=(x-1)*T(x,n-1).
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EXAMPLE
| {1},
{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}
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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]
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CROSSREFS
| Cf. A053120.
Sequence in context: A071426 A135356 A191362 * A139139 A077872 A094053
Adjacent sequences: A137419 A137420 A137421 * A137423 A137424 A137425
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Apr 16 2008
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