A136664 Triangular vector sequence as weighted conversion between A137286 and A049310.

1, 0, 2, 8, 0, 4, 0, 20, 0, 8, 128, 0, 48, 0, 16, 0, 352, 0, 112, 0, 32, 3072, 0, 928, 0, 256, 0, 64, 0, 8928, 0, 2368, 0, 576, 0, 128, 98304, 0, 24960, 0, 5888, 0, 1280, 0, 256, 0, 296448, 0, 67584, 0, 14336, 0, 2816, 0, 512, 3932160, 0, 863232, 0, 178176, 0, 34304

Offset

1,3

Comments

Row sums:

{1, 2, 12, 28, 192, 496, 4320, 12000, 130688, 381696, 5015040};

Suppose that you have a Chebyshev-like recursion: (one type) P[x,n]=x*P[x,n-1]-P[x,n-2]

and an Hermite: Q[x,n]=x*Q[x,n-1]-n*Q[x,n-2]

You can define a set of Matrices on the Coefficient list vectors:

vp[n]=M[n].vq[n]

vq[n].vq[n]t=delta[i,j]

vp[n].vq[n]t=M[n]

where M[n] is a diagonal matrix (a vector)

Then a new set of polynomials is obtained.

Links

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

Formula

T(n,m)=If[A137286(m)>0,A049310(n)/A137286(m),0] Out_vector=2^(n-1)*T(n,m)

Example

{1},
{0, 2},
{8, 0, 4},
{0, 20, 0, 8},
{128, 0, 48, 0, 16},
{0, 352, 0, 112, 0, 32},
{3072, 0, 928, 0, 256, 0, 64},
{0, 8928, 0, 2368, 0, 576, 0, 128},
{98304, 0, 24960, 0, 5888, 0, 1280, 0, 256},
{0, 296448, 0, 67584, 0, 14336, 0, 2816, 0, 512},
{3932160, 0, 863232, 0, 178176, 0, 34304, 0, 6144, 0, 1024}

Mathematica

Clear[P, x, n, a] (*Hermite : A137286*) P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; a1 = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; (* Chebyshev : other kind : A049310*) Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = 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, 2^(n - 1)*a1[[n]][[i]]/a[[n]][[i]]], {i, 1, Length[a[[n]]]}], {n, 1, Length[a]}]; Flatten[b]

Crossrefs

Cf. A137286, A049310.

Sequence in context: A197252 A244122 A021785 * A086728 A118292 A160584

Adjacent sequences: A136661 A136662 A136663 * A136665 A136666 A136667

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Apr 01 2008

Status

approved