login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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, 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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,uned,tabl
AUTHOR
Roger L. Bagula, Apr 02 2008
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 21 05:50 EDT 2024. Contains 373540 sequences. (Running on oeis4.)