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A138090 A triangular sequence of three 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). 0

%I #3 Mar 30 2012 17:34:26

%S 1,0,2,-2,0,4,0,-10,0,12,8,0,-36,0,32,0,66,0,-140,0,80,-48,0,348,0,

%T -512,0,208,0,-558,0,1708,0,-1728,0,544,384,0,-3900,0,7776,0,-5680,0,

%U 1408,0,5790,0,-23364,0,32496,0,-18304,0,3648,-3840,0,50580,0,-126720,0,128624,0,-57600,0,9472

%N A triangular sequence of three 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).

%C Row sums are:

%C {1, 2, 2, 2, 4, 6, -4, -34, -12, 266, 516};

%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 cubic fields and third derivatives.

%F P(x, n) = 2*x*P(x, n - 1) - n*P(x, n - 2) + 4*x^3*P(x, n - 3); 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, 66, 0, -140, 0, 80},

%e {-48, 0, 348, 0, -512, 0, 208},

%e {0, -558, 0, 1708, 0, -1728, 0, 544},

%e {384, 0, -3900, 0, 7776, 0, -5680, 0, 1408},

%e {0, 5790, 0, -23364, 0, 32496, 0, -18304, 0,3648},

%e {-3840, 0, 50580, 0, -126720, 0, 128624, 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]; 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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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)