A136666 triangle of conversion vectors/ coefficients between adjusted to be Integers: Hermite-like: p(x,n)=2*x*p(x,n-1)-n*p(x,n-1); and Chebyshev-like: q(x,n)=2*x*q(x,n-1)-q(x,n-1);.

0, 0, 2, 16, 0, 8, 0, 56, 0, 24, 512, 0, 176, 0, 64, 0, 1632, 0, 512, 0, 160, 18432, 0, 4992, 0, 1408, 0, 384, 0, 57984, 0, 14656, 0, 3712, 0, 896, 786432, 0, 177792, 0, 41472, 0, 9472, 0, 2048, 0, 2480640, 0, 529920, 0, 113664, 0, 23552, 0, 4608, 39321600, 0

Offset

1,3

Comments

Row sums are:

{0, 2, 24, 80, 752, 2304, 25216, 77248, 1017216, 3152384, 48902144}

The results appear to be alternating orthogonal by the double integral

test using the Hermite weight function:

b = Table[Apply[Plus, Table[If[a[[n]][[i]] == 0, 0, x^(i - 1)*a1[[n]][[i]]/a[[n]][[i]]], {i, 1, Length[a[[n]]]}]], {n, 1, Length[a]}];

Table[Integrate[Exp[ -x^2/2]*b[[n]]*b[[m]], {x, -Infinity, Infinity}], {n, 1, Length[b]}, {m, 1, Length[b]}]

Links

Table of n, a(n) for n=1..57.

Formula

vector t(n)=2^n*n*If[ Coefficient[p[x,n]]<.0,ratio of coefficients of q(x,n)/p(x,n)]

Example

{0},
{0, 2},
{16, 0, 8},
{0, 56, 0, 24},
{512, 0, 176, 0, 64},
{0, 1632, 0, 512, 0, 160},
{18432, 0, 4992, 0, 1408, 0, 384},
{0, 57984, 0, 14656, 0, 3712, 0, 896},
{786432, 0, 177792, 0, 41472, 0, 9472, 0, 2048},
{0, 2480640, 0, 529920, 0, 113664, 0, 23552, 0, 4608},
{39321600, 0, 7673856, 0, 1536000, 0, 303104, 0, 57344, 0, 10240}

Mathematica

Clear[P, x, n, a] (*Hermite-like*) P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - n*P[x, n - 2]; a1 = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; (* Chebyshev-like*) Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = 2*x*B[x, n - 1] - B[x, n - 2]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; (* converter*) b = Table[Table[If[a[[n]][[i]] == 0, 0, (n - 1)*2^(n - 1)* a1[[n]][[i]]/a[[n]][[i]]], {i, 1, Length[a[[n]]]}], {n, 1, Length[a]}]; Flatten[b]

Crossrefs

Sequence in context: A056072 A294195 A175982 * A031361 A155955 A221076

Adjacent sequences: A136663 A136664 A136665 * A136667 A136668 A136669

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Apr 02 2008

Status

approved