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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

Table of n, a(n) for n=1..65.

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.)