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.
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
KEYWORD
AUTHOR
Roger L. Bagula, May 02 2008
STATUS
approved