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).

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, -512, 0, 208, 0, -558, 0, 1708, 0, -1728, 0, 544, 384, 0, -3900, 0, 7776, 0, -5680, 0, 1408, 0, 5790, 0, -23364, 0, 32496, 0, -18304, 0, 3648, -3840, 0, 50580, 0, -126720, 0, 128624, 0, -57600, 0, 9472

Offset

1,3

Comments

Row sums are:

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

The alternating orthogonal integration is:

Table[Integrate[P[x, n]*P[x, m]*Exp[ -x^2/2], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0,10}] // TableForm;

This sequence is the result of a thought experiment for cubic fields and third derivatives.

Links

Table of n, a(n) for n=1..66.

Formula

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)).

Example

{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, -512, 0, 208},
{0, -558, 0, 1708, 0, -1728, 0, 544},
{384, 0, -3900, 0, 7776, 0, -5680, 0, 1408},
{0, 5790, 0, -23364, 0, 32496, 0, -18304, 0,3648},
{-3840, 0, 50580, 0, -126720, 0, 128624, 0, -57600, 0, 9472}

Mathematica

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}];

Crossrefs

Sequence in context: A112080 A052176 A138092 * A138093 A138094 A060821

Adjacent sequences: A138087 A138088 A138089 * A138091 A138092 A138093

Keyword

uned,tabl,sign

Author

Roger L. Bagula, May 02 2008

Status

approved