It’s still not clear to me what you are looking for. You say
GetCoeff, that takes an integer n, a power series z and a monomial x^n,
But, the function you posted
GetCoeff takes two arguments, not three; and they both appear to be integers, rather than a power series or monomial. You refer once to
x^n. You refer to a fixed polynomial z + z^2 + z^3. The function
GetCoeff returns a polynomial in
z[i] for i \in [0,1,...n]. Do these refer to the same z ? Perhaps
i represents integer-valued t ? It would help if you explain things more precisely.
In general, these algebra tools involve a trade-off between flexibility-expressiveness on one hand and efficiency on the other. As you noted, Mathematica is more expressive than efficient. (Although a few decades and a few hundred million dollars can go a long way to closing this gap.) I’m not sure, but it looks like Nemo.jl and AbstractAlgebra.jl can’t do what you want easily; they want numeric coefficients. But, maybe asking the developers is worthwhile. There is also JuliaAlgebra, as well as a few other polynomial-related packages.
@dpsanders suggested TaylorSeries.jl, which looks like a more likely candidate than the others.
Following is your Mathematica code translated into Symata.jl. But, I don’t have access to Mathematica at the moment, so I can’t test it.
GetCoeff(p_, n_) := GetCoeff(p, n) = Expand(If(
p == 1, z(n),
If(p == 2, Sum(z(i) * z(n - i), [i, 0, n]),
Apply(Plus, Table(GetCoeff(p - 1, i) * z(n - i), [i, 0, n])))))
symata 12> Get("getcoeff.sj")
symata 12> GetCoeff(3,5)
Out(12) = 7z(0)*z(2)*z(3) + 7z(0)*z(1)*z(4) + 7*z(0)^2*z(5)
symata 13> ? GetCoeff
GetCoeff(3,5)=(7z(0)*z(2)*z(3) + 7z(0)*z(1)*z(4) + 7*z(0)^2*z(5))
GetCoeff(p_,n_):=GetCoeff(p,n)=CompoundExpression(nothing,Expand(If(p == 1,z(n),If(p == 2,Sum((z(i)*z(n - i)),[i,0,n]),Apply(Plus,Table((GetCoeff((p - 1),i)*z(n - i)),[i,0,n]))))))
You can probably gain some efficiency by using
Symata from the
Julia side so that you can control the evaluation. Or, you can work directly in SymPy.jl. Or maybe SymEngine.jl, but it is not well-developed compared to