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Search: id:A137448
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| A137448 |
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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 is differentiated twice : p(x,n)=-x^2*T''(x,n)+T(x,n). |
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+0
1
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| 1, 1, 1, -1, 0, -2, 0, -3, 0, -20, 1, 0, 8, 0, -88, 0, 5, 0, 100, 0, -304, -1, 0, -18, 0, 528, 0, -928, 0, -7, 0, -280, 0, 2128, 0, -2624, 1, 0, 32, 0, -1760, 0, 7424, 0, -7040, 0, 9, 0, 600, 0, -8208, 0, 23616, 0, -18176, -1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568
(list; table; graph; listen)
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OFFSET
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1,6
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COMMENT
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Row sums are:
{1, 2, -3, -23, -79, -199, -419, -783, -1343, -2159, -3299};
As an operator algebra like an Energy Hamiltonian:
e(n)*T(x,n)=p(x,n)/x^2
Solving at the Chebyshev border of x=1 gives the row sums:
Table[e /. Solve[e*ChebyshevT[n, x] - p[x, n] == 0, e], {n, 0, 10}] /. x -> 1
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FORMULA
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p(x,n)=-x^2*T''(x,n)+T(x,n)
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EXAMPLE
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{1},
{1, 1},
{-1, 0, -2},
{0, -3, 0, -20},
{1, 0, 8, 0, -88},
{0, 5, 0, 100, 0, -304},
{-1, 0, -18, 0, 528, 0, -928},
{0, -7, 0, -280, 0, 2128, 0, -2624},
{1, 0, 32, 0, -1760, 0, 7424, 0, -7040},
{0, 9, 0, 600, 0, -8208,0, 23616, 0, -18176},
{-1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568}
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MATHEMATICA
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Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[ChebyshevT[n, x], {x, 2}] + ChebyshevT[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Sequence in context: A067165 A079981 A117776 this_sequence A035165 A079133 A143143
Adjacent sequences: A137445 A137446 A137447 this_sequence A137449 A137450 A137451
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Apr 18 2008
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