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A140698
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Triangular sequence of anti-diagonal series of Galois GF(2^Prime[n]) polynomials to Cyclotomic polynomial: Galois polynomial GF(2^p) g[x,p]=x^p+x+1 Cyclotomic polynomial for primes: c[x,p]=Sum[x^i,{i,0,p}] ratio polynomial: q[x,p]=c[x,p]/g[x,p] Toral inverse for expansion: p[x,p]=x^p*g[1/x,p]/(x^p*c[x,p]).
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1, 0, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 0, -1, 1, -1, -2, 0, 0, -1, 1, 1, 1, 1, 0, 0, -1, 1, -1, 1, 1, 0, 0, 0, -1, 1, 1, -2, -2, 0, 0, 0, 0, -1, 1, -1, 1, 0, 1, 0, 0, 0, 0, -1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,17
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COMMENTS
| Row sums are;
{1, 1, 1, 0, 2, -3, 3, 1, -3, 1, 3};
The new polynomials that result should be "Field -Like"
as well as they are representation of the quotient group of the type;
GF(2^Prime[n])/Cyclotomic(Prime[n])
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FORMULA
| Galois polynomial GF(2^p) g[x,p]=x^p+x+1 Cyclotomic polynomial for primes: c[x,p]=Sum[x^i,{i,0,p}] ratio polynomial: q[x,p]=c[x,p]/g[x,p] Toral inverse for expansion: p[x,p]=x^p*g[1/x,p]/(x^p*c[x,p]) a(n,m)=Anti-diagonal Coefficients(p[x,p]).
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EXAMPLE
| {{1},
{0, 1},
{1, -1, 1},
{-1, 1, -1, 1},
{1, 1, 0, -1, 1},
{-1, -2, 0, 0, -1, 1},
{1, 1, 1, 0, 0, -1, 1},
{-1, 1, 1, 0, 0, 0, -1, 1},
{1, -2, -2, 0, 0, 0, 0, -1, 1},
{-1, 1, 0, 1, 0, 0, 0, 0, -1, 1},
{1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1}}
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MATHEMATICA
| p[x_, n_] = (x^Prime[n] + x^(Prime[n] - 1) + 1)/Cyclotomic[Prime[n], x] a = Table[CoefficientList[Normal[Series[p[x, n], {x, 0, 30}]], x], {n, 1, 31}]; (* anti-diagonal triangular sequence representation*) b = Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 11}]; Flatten[b]
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CROSSREFS
| Sequence in context: A060952 A037844 A037880 * A124764 A151899 A204263
Adjacent sequences: A140695 A140696 A140697 * A140699 A140700 A140701
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 11 2008
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