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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
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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; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| 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
| 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
| 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
| {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
| 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
| Sequence in context: A117406 A151844 A008783 * A081576 A054654 A154477
Adjacent sequences: A139141 A139142 A139143 * A139145 A139146 A139147
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 05 2008
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