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A138094 A triangular sequence of eight back recursive polynomials that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]:k=8 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
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, 624, 10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680, 0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512, 188320, 0, 459736, 0, 382592, 0, 198688 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums are:

{1, 2, 6, 22, 92, 374, 1832, 8458, 46272, 236048, 1306268};

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 8th derivatives.

The lower 7 row sums are the same as k=6: only

the higher values are really different.

LINKS

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

FORMULA

k=8 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, 624},

{10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680},

{0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512},

{188320, 0, 459736, 0, 382592, 0, 198688, 0, 64800, 0, 12132}

MATHEMATICA

Clear[P, x]:k=8; 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 A138093.

Sequence in context: A138092 A138090 A138093 * A060821 A191718 A286777

Adjacent sequences:  A138091 A138092 A138093 * A138095 A138096 A138097

KEYWORD

nonn,uned,tabl

AUTHOR

Roger L. Bagula, May 02 2008

STATUS

approved

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)