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Search: id:A139144
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| A139144 |
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Triangular sequence of coefficients of central statistical moments as a recursion. c = -(x - x^2); b = (-1 - a + 2 x)/x; a = 0; p(x, n) = (a + b*x)*p(x, n - 1) + c*p(x, n - 2}. |
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+0
1
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| 1, 0, 0, 1, -1, 0, 1, -3, 2, 0, 1, -4, 6, -3, 0, 1, -5, 10, -10, 4, 0, -1, 6, -15, 20, -15, 5, 0, 1, -9, 33, -65, 75, -49, 14, 0, -1, 12, -58, 152, -240, 234, -132, 33, 0, 1, -15, 92, -310, 642, -854, 724, -360, 80, 0, -1, 18, -135, 564, -1472, 2530, -2906, 2174, -965, 193
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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Row sums: {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...};
The p(x,0) to p(x,5) are from the MathWorld page and p(x,6) to p(x,10) are recursively generated.
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REFERENCES
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Charles D. Hodgeman, ed., "CRC Standard Mathematical Tables and Formulae", 12th Edition, page 391
Samuel M. Selby, ed., "CRC Standard Mathematical Tables and Formulae",16th Edition, page 530
Weisstein, Eric W. "Central Moment." http://mathworld.wolfram.com/CentralMoment.html
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FORMULA
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c = -(x - x^2); b = (-1 - a + 2 x)/x; a = 0; p(x, n) = (a + b*x)*p(x, n - 1) + c*p(x, n - 2}; out_n,m=Coefficients(p(x,n)).
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EXAMPLE
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{1},
{0},
{0, 1, -1},
{0, 1, -3, 2},
{0, 1, -4, 6, -3},
{0, 1, -5, 10, -10, 4},
{0, -1,6, -15, 20, -15, 5},
{0, 1, -9, 33, -65, 75, -49, 14},
{0, -1, 12, -58, 152, -240, 234, -132, 33},
{0, 1, -15, 92, -310, 642, -854, 724, -360, 80},
{0, -1, 18, -135, 564, -1472, 2530, -2906, 2174, -965, 193}
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MATHEMATICA
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Clear[p, x, a] p[x, 0] = 1; p[x, 1] = 0; p[x, 2] = -x^2 + x; p[x, 3] = 2*x^3 - 3*x^2 + x; p[x, 4] = -3*x^4 + 6*x^3 - 4*x^2 + x; p[x, 5] = 4*x^5 - 10*x^4 + 10*x^3 - 5*x^2 + x; c = -(x - x^2); b = (-1 - a + 2 x)/x; a = 0; p[x_, n_] := p[x, n] = (a + b*x)*p[x, n - 1] + c*p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a0] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]
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CROSSREFS
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Adjacent sequences: A139141 A139142 A139143 this_sequence A139145 A139146 A139147
Sequence in context: A117406 A151844 A008783 this_sequence A081576 A054654 A154477
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 05 2008
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