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.
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
KEYWORD
AUTHOR
Roger L. Bagula, Apr 01 2008
STATUS
approved