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A136664 Triangular vector sequence as weighted conversion between A137286 and A049310. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified March 28 10:32 EDT 2020. Contains 333083 sequences. (Running on oeis4.)