1,2

Here "generic" means that no coefficient in the polynomial is zero. - Artur Jasinski (grafix(AT)csl.pl), Nov 01 2007

At one point it was suggested that this is the same sequence as A039744, but this is wrong. Dean Hickerson, Dec 16 2006, comments as follows: (Start)

The claim that A039744 equals the number of monomials in the discriminant is false. The first counterexample is n=4: There are 18 such partitions, but the discriminant has no terms corresponding to the partitions 3+2+2+2+2+1 and 2+2+2+2+2+2, so the number of monomials in the discriminant is only 16.

According to Wikipedia, the discriminant of a_0 + a_1 x + ... + a_n x^n is 1/a_n times the determinant of a particular matrix; for n=4 that matrix is

[ a_4...a_3...a_2...a_1...a_0...0.....0... ]

[ 0.....a_4...a_3...a_2...a_1...a_0...0... ]

[ 0.....0.....a_4...a_3...a_2...a_1...a_0. ]

[ 4a_4..3a_3..2a_2..1a_1..0.....0.....0... ]

[ 0.....4a_4..3a_3..2a_2..1a_1..0.....0... ]

[ 0.....0.....4a_4..3a_3..2a_2..1a_1..0... ]

[ 0.....0.....0.....4a_4..3a_3..2a_2..1a_1 ]

It is easy to see that there are no monomials in the expansion of this involving either a_4 * a_3 * a_2^4 * a_1 or a_4 * a_2^6.

For larger n, it's not clear to me what restrictions need to be put on the partitions to guarantee that the corresponding monomials occur in the expansion of the determinant.

Columns near the left or right have very few nonzero elements and this adds some restrictions to the partitions.

For example, from column 2 of the matrix, we see that the partition must have at least one term equal to n or n-1. From the last column, it must have at least one term equal to 0 or 1. Maybe the complete list of such conditions is enough; I don't know.

Even if we could figure out exactly which partitions correspond to monomials that occur in the expansion, I can't rule out the possibility that the coefficients of some such monomial could cancel out, further reducing the number of nonzero monomials in the discriminant. (End)

Discriminant of cubic K3*x^3 + K2*x^2 + K1*x + K0 is -27*K3^2*K0^2 + 18*K3*K2*K1*K0 - 4*K2^3*K0 - 4*K3*K1^3 + K2^2*K1^2 which contains 5 monomials. - Bill Daly (bill.daly(AT)tradition.co.uk)

A007878 := proc(n) local x, a, ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), x)) end;

Clear[f, g]; g[0] = f[0]; g[n_Integer?Positive] := g[n] = g[n - 1] + f[n] x^n; myFun[n_Integer?Positive] := Length@Resultant[g[n], D[g[n], x], x, Method -> "BezoutMatrix"]; Table[myFun[n], {n, 1, 8}] (* Procedure from Artur Jasinski, improved by Jean-Marc Gulliet (jeanmarc.gulliet(AT)gmail.com) *)

(MAGMA) 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 ] ] - from Victor S. Miller, Dec 16 2006

Adjacent sequences: A007875 A007876 A007877 this_sequence A007879 A007880 A007881

Sequence in context: A009225 A019448 A000753 this_sequence A019589 A087949 A028333

nonn,nice,hard,more

reiner(AT)math.umn.edu

9th term from Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Entry revised by njas, Dec 16 2006

a(10) from Artur Jasinski (grafix(AT)csl.pl), Apr 02 2008