A140666 - OEIS

%I #7 Aug 05 2019 19:26:03

%S 1,-1,2,2,0,2,2,0,2,0,2,2,0,2,0,2,0,2,0,2,2,0,2,0,2,0,2,0,2,0,2,2,0,2,

%T 0,2,0,2,0,2,0,2,0,2,0,2,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,2,0,2,0,2,

%U 0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2

%N A triangle of coefficients of the difference of prime cyclotomic doubled polynomials: p(x,n)=(c(x,Prime[n])-c(x,2*Prime[n]))/x.

%C Row sums are: {0, 2, 4, 6, 10, 12, 16, 18, 22, 28, ...}

%C The factor x is used instead of 2x to get an integer n=1 term.

%C p(x,n)/2 are related to the double product:two primes n,m such that

%C Cyclotomic[Prime[n], x]* Cyclotomic[2*Prime[n]=(Cyclotomic[Prime[m], x] - Cyclotomic[2*Prime[m], x])/(2*x)

%F c(x,n)=CyclotomicPolynomial; c(x,Prime[n])=(x^Prime[n]-1)/(x-1); p(x,n)=(c(x,Prime[n])-c(x,2*Prime[n]))/x; a(n,m)=Coefficients(p(x,n))

%e {1, -1},

%e {2},

%e {2, 0, 2},

%e {2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2},

%e {2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2}

%t Clear[p, x, n] p[x_, n_] = (Cyclotomic[Prime[n], x] - Cyclotomic[2*Prime[n], x])/x; Table[ExpandAll[p[x, n]], {n, 1, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[a]

%K tabf,uned,sign

%O 1,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jul 11 2008