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 A139144 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}. 0
 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; text; internal format)
 OFFSET 1,8 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. 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 LINKS 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)). 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} 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}] CROSSREFS Sequence in context: A151844 A286223 A008783 * A081576 A292717 A054654 Adjacent sequences:  A139141 A139142 A139143 * A139145 A139146 A139147 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Jun 05 2008 STATUS approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)