Search: id:A139144
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%I A139144
%S A139144 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,
%T A139144 75,49,14,0,1,12,58,152,240,234,132,33,0,1,15,92,310,642,854,724,360,80,
%U A139144 0,1,18,135,564,1472,2530,2906,2174,965,193
%V A139144 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,
%W A139144 -49,14,0,-1,12,-58,152,-240,234,-132,33,0,1,-15,92,-310,642,-854,724,
               -360,80,0,-1,18,
%X A139144 -135,564,-1472,2530,-2906,2174,-965,193
%N 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}.
%C A139144 Row sums: {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...};
%C A139144 The p(x,0) to p(x,5) are from the MathWorld page and p(x,6) to p(x,10) 
               are recursively generated.
%D A139144 Charles D. Hodgeman, ed., "CRC Standard Mathematical Tables and Formulae", 
               12th Edition, page 391
%D A139144 Samuel M. Selby, ed., "CRC Standard Mathematical Tables and Formulae",
               16th Edition, page 530
%D A139144 Weisstein, Eric W. "Central Moment." http://mathworld.wolfram.com/CentralMoment.html
%F A139144 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)).
%e A139144 {1},
%e A139144 {0},
%e A139144 {0, 1, -1},
%e A139144 {0, 1, -3, 2},
%e A139144 {0, 1, -4, 6, -3},
%e A139144 {0, 1, -5, 10, -10, 4},
%e A139144 {0, -1,6, -15, 20, -15, 5},
%e A139144 {0, 1, -9, 33, -65, 75, -49, 14},
%e A139144 {0, -1, 12, -58, 152, -240, 234, -132, 33},
%e A139144 {0, 1, -15, 92, -310, 642, -854, 724, -360, 80},
%e A139144 {0, -1, 18, -135, 564, -1472, 2530, -2906, 2174, -965, 193}
%t A139144 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}]
%Y A139144 Sequence in context: A117406 A151844 A008783 this_sequence A081576 A054654 
               A154477
%Y A139144 Adjacent sequences: A139141 A139142 A139143 this_sequence A139145 A139146 
               A139147
%K A139144 uned,tabl,sign
%O A139144 1,8
%A A139144 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 05 2008

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