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%I #3 Mar 30 2012 17:34:26
%S 1,0,2,-2,0,4,0,-10,0,12,-8,0,-36,0,32,0,-16,0,-140,0,80,120,0,40,0,
%T -512,0,208,0,842,0,440,0,-1728,0,544,-448,0,3668,0,2864,0,-5680,0,
%U 1408,0,-7178,0,15196,0,14960,0,-18304,0,3648,-7520,0,-55036,0,56320,0,67680,0,-57600,0,9472
%N A triangular sequence of four back recursive polynomial that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]: P(x, n) = 2*x*P(x, n - 1) - n*P(x, n - 2) + 4*x^3*P(x, n - 3)-n^2*P(x, n - 4).
%C Row sums are:
%C {1, 2, 2, 2, -12, -76, -144, 98, 1812, 8322, 13316};
%C The alternating orthogonal integration is:
%C Table[Integrate[P[x, n]*P[x, m]*Exp[ -x^2/2], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0,10}] // TableForm;
%C This sequence is the result of a thought experiment for fourth derivatives.
%F P(x, n) = 2*x*P(x, n - 1) - n*P(x, n - 2) + 4*x^3*P(x, n - 3)-n^2*P(x, n - 4); out_n,m=Coefficients(P(x,n)).
%e {1},
%e {0, 2},
%e {-2, 0, 4},
%e {0, -10, 0, 12},
%e {-8, 0, -36, 0, 32},
%e {0, -16, 0, -140, 0, 80},
%e {120, 0, 40, 0, -512, 0, 208},
%e {0, 842, 0, 440, 0, -1728, 0, 544},
%e {-448, 0, 3668, 0, 2864, 0, -5680, 0, 1408},
%e {0, -7178, 0, 15196, 0, 14960, 0, -18304, 0, 3648},
%e {-7520, 0, -55036, 0, 56320, 0, 67680, 0, -57600, 0, 9472}
%t Clear[P, x] P[x, -2] = 0; P[x, -1] = 0; P[x, 0] = 1; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - n*P[x, n - 2] + 4*x^3*P[x, n - 3]-n^2*P[x, n - 4]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];
%K uned,tabl,sign
%O 1,3
%A _Roger L. Bagula_, May 02 2008
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