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A355511
a(n) is the number of monic polynomials of degree n over GF(11) without linear factors.
0
0, 0, 55, 440, 5170, 56408, 620950, 6830120, 75131485, 826446280, 9090909091, 100000000000, 1100000000000, 12100000000000, 133100000000000, 1464100000000000, 16105100000000000, 177156100000000000, 1948717100000000000, 21435888100000000000, 235794769100000000000
OFFSET
0,3
FORMULA
O.g.f. (1 - z)^(11)/(1-11*z) - 1
EXAMPLE
a(0) = 0 since there are no irreducible constant polynomials (as GF(11) is a field).
a(1) = 0 since all polynomials of degree 1 have linear factors.
a(2), the number of quadratic polynomials without linear factors, then coincides with the number of irreducible quadratics in GF(11), which is known to be M(11,2), where M(a,d) is the necklace polynomial, so a(2) = 55.
MATHEMATICA
necklacePolynomial[q_, n_] :=
necklacePolynomial[q, n] = (1/n)*
DivisorSum[n, MoebiusMu[n/#1]*q^#1 & ];
numIrreds[q_, n_] := If[n != 0, necklacePolynomial[q, n], 0];
restrictedPolynomialsOGF[q_, n_, d_] :=
Product[(1 - z^If[ArrayDepth[d[[l]]] == 0, d[[l]], d[[l]][[1]]])^
If[ArrayDepth[d[[l]]] == 0, numIrreds[q, d[[l]]],
d[[l]][[2]]], {l, 1, Length[d]}]/(1 - q*z);
numRestrictedPolys[q_, n_, d_] :=
SeriesCoefficient[restrictedPolynomialsOGF[q, n, d], {z, 0, n}];
q = 11;
TableForm[{#, numRestrictedPolys[q, #, {1}]} & /@ (Range[20]),
TableHeadings -> {{Row[{"(q=", q, ")"}]}, {"n", "#rootless monics"}}]
CROSSREFS
Cf. A355510.
Sequence in context: A222348 A075740 A340240 * A129217 A116060 A145054
KEYWORD
nonn
AUTHOR
Greyson C. Wesley, Jul 04 2022
STATUS
approved