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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..20. 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 Adjacent sequences: A355508 A355509 A355510 * A355512 A355513 A355514 KEYWORD nonn AUTHOR Greyson C. Wesley, Jul 04 2022 STATUS approved

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Last modified September 8 21:48 EDT 2024. Contains 375759 sequences. (Running on oeis4.)