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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).

%I A139583 #9 Aug 12 2019 18:33:58
%S A139583 3,2,4,-2,0,8,-4,-24,0,16,4,0,-96,0,32,-8,240,0,-320,0,64,-56,0,1440,
%T A139583 0,-960,0,128,464,-3360,0,6720,0,-2688,0,256,1712,0,-26880,0,26880,0,
%U A139583 -7168,0,512,-10720,60480,0,-161280,0,96768,0,-18432,0,1024,-52256,0,604800,0,-806400,0,322560,0,-46080,0,2048
%N 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).
%C A139583 Row sums are:
%C A139583 {3, 6, 6, -12, -60, -24, 552, 1392, -4944, -32160, 24672}
%C A139583 Implicit surface of:
%C A139583 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
%C A139583 has twelve horns off the main body ( more than the Chebyshev at the same level).
%F A139583 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).
%e A139583 {3},
%e A139583 {2, 4},
%e A139583 {-2, 0, 8},
%e A139583 {-4, -24, 0, 16},
%e A139583 {4, 0, -96, 0, 32},
%e A139583 {-8, 240, 0, -320, 0, 64},
%e A139583 {-56, 0,1440, 0, -960, 0, 128},
%e A139583 {464, -3360, 0, 6720, 0, -2688, 0, 256},
%e A139583 {1712, 0, -26880, 0, 26880, 0, -7168, 0, 512},
%e A139583 {-10720, 60480, 0, -161280, 0, 96768, 0, -18432, 0, 1024},
%e A139583 {-52256, 0, 604800, 0, -806400, 0, 322560, 0, -46080, 0, 2048}
%t A139583 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]
%Y A139583 Cf. A060821.
%K A139583 uned,sign
%O A139583 1,1
%A A139583 _Roger L. Bagula_ and _Gary W. Adamson_, Jun 11 2008