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

Author

Roger L. Bagula, Apr 01 2008

Keywords

• nonn
• uned
• tabl

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.

Examples

			{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}
		

Crossrefs

Cf. A137286, A049310.

Programs

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]

Formula

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