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A140698 Triangular sequence of antidiagonal 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]). 0
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; text; internal format)
OFFSET

1,17

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))).

LINKS

Table of n, a(n) for n=1..66.

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)=Antidiagonal Coefficients(p[x,p]).

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}}

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}]; (* antidiagonal triangular sequence representation *) b = Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 11}]; Flatten[b]

CROSSREFS

Sequence in context: A037844 A037880 A241035 * A231599 A321924 A124764

Adjacent sequences:  A140695 A140696 A140697 * A140699 A140700 A140701

KEYWORD

uned,tabl,sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Jul 11 2008

STATUS

approved

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)