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A137350
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A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=x*p(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).
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0
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1, -1, 1, 0, 1, -1, -1, 1, 1, -1, 1, 0, -2, -1, 1, 1, 2, -2, 1, -1, 1, -3, -1, 1, 0, 3, 3, -3, 1, -1, -3, 3, -4, -1, 1, 1, -1, 6, 4, -4, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,13
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COMMENTS
| Row sums are:
{1, 0, 1, -1, 1, -2, 2, -3, 4, -5, 7}
In differential equation terms this is equivalent to ( in Mathematica notation):
D[y[x],{x,3}]=x*D[y[x],{x,1}]-y[x];
Two simple possible HypergeometricPFQ based results are:
DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 1}, y, x];
DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 0}, y, x].
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FORMULA
| Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).
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EXAMPLE
| {1},
{-1, 1},
{0, 1},
{-1, -1, 1},
{1, -1, 1},
{0, -2, -1, 1},
{1, 2, -2, 1},
{-1, 1, -3, -1, 1},
{0, 3, 3, -3, 1},
{-1, -3, 3, -4, -1, 1},
{1, -1, 6, 4, -4, 1}
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MATHEMATICA
| Clear[Q, x] Q[x, -2] = 1 - x; Q[x, -1] = 0; Q[x, 0] = 1; Q[x_, n_] := Q[x, n] = x*Q[x, n - 2] - Q[x, n - 3]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
| Cf. A000931, A137276.
Sequence in context: A055215 A058398 A091499 * A166240 A114087 A008284
Adjacent sequences: A137347 A137348 A137349 * A137351 A137352 A137353
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 08 2008
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