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A140666
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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.
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0
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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, 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, 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
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OFFSET
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1,3
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COMMENTS
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Row sums are: {0, 2, 4, 6, 10, 12, 16, 18, 22, 28, ...}
The factor x is used instead of 2x to get an integer n=1 term.
p(x,n)/2 are related to the double product:two primes n,m such that
Cyclotomic[Prime[n], x]* Cyclotomic[2*Prime[n]=(Cyclotomic[Prime[m], x] - Cyclotomic[2*Prime[m], x])/(2*x)
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LINKS
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FORMULA
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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))
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EXAMPLE
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{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, 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, 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}
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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tabf,uned,sign
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AUTHOR
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STATUS
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approved
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