%I A007878
%S A007878 1,2,5,16,59,246,1103,5247,26059,133881
%N A007878 Number of terms in discriminant of generic polynomial of degree n.
%C A007878 Here "generic" means that no coefficient in the polynomial is zero. - 
               Artur Jasinski (grafix(AT)csl.pl), Nov 01 2007
%C A007878 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)
%C A007878 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.
%C A007878 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
%C A007878 [ a_4...a_3...a_2...a_1...a_0...0.....0... ]
%C A007878 [ 0.....a_4...a_3...a_2...a_1...a_0...0... ]
%C A007878 [ 0.....0.....a_4...a_3...a_2...a_1...a_0. ]
%C A007878 [ 4a_4..3a_3..2a_2..1a_1..0.....0.....0... ]
%C A007878 [ 0.....4a_4..3a_3..2a_2..1a_1..0.....0... ]
%C A007878 [ 0.....0.....4a_4..3a_3..2a_2..1a_1..0... ]
%C A007878 [ 0.....0.....0.....4a_4..3a_3..2a_2..1a_1 ]
%C A007878 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.
%C A007878 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.
%C A007878 Columns near the left or right have very few nonzero elements and this 
               adds some restrictions to the partitions.
%C A007878 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.
%C A007878 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)
%e A007878 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)
%p A007878 A007878 := proc(n) local x,a,ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), 
               x)) end;
%t A007878 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) 
               *)
%o A007878 (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
%Y A007878 Sequence in context: A149979 A019448 A000753 this_sequence A019589 A087949 
               A028333
%Y A007878 Adjacent sequences: A007875 A007876 A007877 this_sequence A007879 A007880 
               A007881
%K A007878 nonn,nice,hard,more
%O A007878 1,2
%A A007878 reiner(AT)math.umn.edu
%E A007878 9th term from Lyle Ramshaw (ramshaw(AT)pa.dec.com)
%E A007878 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 16 2006
%E A007878 a(10) from Artur Jasinski (grafix(AT)csl.pl), Apr 02 2008