A139583 A triangle of coefficients from Hermite polynomials A060821 as {x,y},{y,z},{z,x} binomials reduced to x: f(x,y,n)=Sum[Coefficients(H(x,n))(i)*x^i*y^(n-1),{i,0,n}]; p(x,y,z)=f(x,y,n)+f(y,z,n)+f(z,x,n).

3, 2, 4, -2, 0, 8, -4, -24, 0, 16, 4, 0, -96, 0, 32, -8, 240, 0, -320, 0, 64, -56, 0, 1440, 0, -960, 0, 128, 464, -3360, 0, 6720, 0, -2688, 0, 256, 1712, 0, -26880, 0, 26880, 0, -7168, 0, 512, -10720, 60480, 0, -161280, 0, 96768, 0, -18432, 0, 1024, -52256, 0, 604800, 0, -806400, 0, 322560, 0, -46080, 0, 2048

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Jun 11 2008

Keywords

• uned
• sign

Comments

Row sums are:

{3, 6, 6, -12, -60, -24, 552, 1392, -4944, -32160, 24672}

Implicit surface of:

f[x_, y_, z_] = 128 x^6 - 480 x^4 y^2 + 720 x^2 y^4 - 56 y^6 - 480 x^4 z^2 - 480y^4 z^2 + 720 x^2 z^4 + 720 y^2 z^4 - 240 z^6 - 1

has twelve horns off the main body ( more than the Chebyshev at the same level).

Examples

			{3},
{2, 4},
{-2, 0, 8},
{-4, -24, 0, 16},
{4, 0, -96, 0, 32},
{-8, 240, 0, -320, 0, 64},
{-56, 0,1440, 0, -960, 0, 128},
{464, -3360, 0, 6720, 0, -2688, 0, 256},
{1712, 0, -26880, 0, 26880, 0, -7168, 0, 512},
{-10720, 60480, 0, -161280, 0, 96768, 0, -18432, 0, 1024},
{-52256, 0, 604800, 0, -806400, 0, 322560, 0, -46080, 0, 2048}
		

Crossrefs

Cf. A060821.

Programs

Mathematica

Clear[f, x, n] f[x_, y_, n_] := Sum[CoefficientList[HermiteH[n, x], x][[i + 1]]*x^i*y^(n - i), {i, 0, Length[CoefficientList[HermiteH[n,x], x]] - 1}]; Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[ x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]

Formula

f(x,y,n)=Sum[Coefficients(H(x,n))(i)*x^i*y^(n-1),{i,0,n}]; p(x,y,z)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(P(x,1,1,n).