A138093 A triangular sequence of six back recursive polynomial that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]:k=6 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}].

1, 0, 2, 2, 0, 4, 0, 10, 0, 12, 24, 0, 36, 0, 32, 0, 148, 0, 140, 0, 86, 432, 0, 656, 0, 512, 0, 232, 0, 3076, 0, 2976, 0, 1782, 0, 616, 6016, 0, 15752, 0, 12688, 0, 6040, 0, 1648, 0, 58994, 0, 80104, 0, 51148, 0, 19888, 0, 4416, 127360, 0, 377108, 0, 382592, 0, 198688

Offset

1,3

Comments

Row sums are:

{1, 2, 6, 22, 92, 374, 1832, 8450, 42144, 214550, 1161784};

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 Sixth derivatives.

Links

Table of n, a(n) for n=1..62.

Formula

P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}]; out_n,m=Coefficients(P(x,n)).

Example

{1},
{0, 2},
{2, 0, 4},
{0, 10, 0, 12},
{24, 0, 36, 0, 32},
{0, 148, 0, 140, 0, 86},
{432, 0, 656, 0, 512, 0, 232},
{0, 3076, 0, 2976, 0, 1782, 0, 616},
{6016, 0, 15752, 0, 12688, 0, 6040, 0, 1648},
{0, 58994, 0, 80104, 0, 51148, 0, 19888, 0, 4416},
{127360, 0, 377108, 0, 382592, 0, 198688, 0, 64224, 0, 11812}

Mathematica

Clear[P, x]:k=6; Table[P[x, -n] = 0, {n, 1, k}]; P[x, 0] = 1; P[x_, n_] := P[x, n] = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P[x, n - m], n^(m/2)*P[x, n - m]], {m, 1, k}];; 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

Similar to but different from A138094.

Sequence in context: A052176 A138092 A138090 * A138094 A060821 A191718

Adjacent sequences: A138090 A138091 A138092 * A138094 A138095 A138096

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, May 02 2008

Status

approved