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A138094
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A triangular sequence of eight back recursive polynomials that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]:k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}].
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1
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1, 0, 2, 2, 0, 4, 0, 10, 0, 12, 24, 0, 36, 0, 32, 0, 148, 0, 140, 0, 86, 432, 0, 656, 0, 512, 0, 232, 0, 3076, 0, 2976, 0, 1782, 0, 624, 10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680, 0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512, 188320, 0, 459736, 0, 382592, 0, 198688
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OFFSET
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1,3
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COMMENTS
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Row sums are:
{1, 2, 6, 22, 92, 374, 1832, 8458, 46272, 236048, 1306268};
The alternating orthogonal integration is:
Table[Integrate[P[x, n]*P[x, m]*Exp[ -x^2/2], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0,10}] // TableForm;
This sequence is the result of a thought experiment for 8th derivatives.
The lower 7 row sums are the same as k=6: only
the higher values are really different.
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LINKS
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FORMULA
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k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}]; out_n,m=Coefficients(P(x,n)).
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EXAMPLE
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{1},
{0, 2},
{2, 0, 4},
{0, 10, 0, 12},
{24, 0, 36, 0, 32},
{0, 148, 0, 140, 0, 86},
{432, 0, 656, 0, 512, 0, 232},
{0, 3076, 0, 2976, 0, 1782, 0, 624},
{10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680},
{0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512},
{188320, 0, 459736, 0, 382592, 0, 198688, 0, 64800, 0, 12132}
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MATHEMATICA
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Clear[P, x]:k=8; Table[P[x, -n] = 0, {n, 1, k}]; P[x, 0] = 1; P[x_, n_] := P[x, n] = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P[x, n - m], n^(m/2)*P[x, n - m]], {m, 1, k}];; 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}];
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CROSSREFS
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Similar to but different from A138093.
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KEYWORD
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AUTHOR
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STATUS
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approved
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