A138092 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).

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, -512, 0, 208, 0, 842, 0, 440, 0, -1728, 0, 544, -448, 0, 3668, 0, 2864, 0, -5680, 0, 1408, 0, -7178, 0, 15196, 0, 14960, 0, -18304, 0, 3648, -7520, 0, -55036, 0, 56320, 0, 67680, 0, -57600, 0, 9472

Offset

1,3

Comments

Row sums are:

{1, 2, 2, 2, -12, -76, -144, 98, 1812, 8322, 13316};

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 fourth 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)-n^2*P(x, n - 4); out_n,m=Coefficients(P(x,n)).

Example

{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, -512, 0, 208},
{0, 842, 0, 440, 0, -1728, 0, 544},
{-448, 0, 3668, 0, 2864, 0, -5680, 0, 1408},
{0, -7178, 0, 15196, 0, 14960, 0, -18304, 0, 3648},
{-7520, 0, -55036, 0, 56320, 0, 67680, 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]-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}];

Crossrefs

Sequence in context: A295215 A112080 A052176 * A138090 A138093 A138094

Adjacent sequences: A138089 A138090 A138091 * A138093 A138094 A138095

Keyword

uned,tabl,sign

Author

Roger L. Bagula, May 02 2008

Status

approved