2nd try: Number of monomials in discriminant of a polynomial of degree n.

[Victor S. Miller]

According to Magma, the sequence starts

foo := > foo := [#Monomials(Numerator(Disc(n))) : n in [1..6]];

[ 1, 2, 5, 16, 59, 246, 1103 ]

I got this withe following program

function Disc(n)
	F := FunctionField(Rationals(),n);
	R<x> := PolynomialRing(F);
	f := x^n + &+[(F.i)*x^(n-i) : i in [1..n]];
	return Discriminant(f);
end function;

[#Monomials(Numerator(Disc(n))) : n in [1..7]]

-- 
Victor S. Miller     | " ... Meanwhile, those of us who can compute can hardly
victor at idaccr.org    | be expected to keep writing papers saying 'I can do the
CCR, Princeton, NJ   | following useless calculation in 2 seconds', and indeed
    08540 USA        | what editor would publish them?"  -- Oliver Atkin