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A355511 a(n) is the number of monic polynomials of degree n over GF(11) without linear factors.

%I A355511 #14 Sep 07 2022 11:12:16
%S A355511 0,0,55,440,5170,56408,620950,6830120,75131485,826446280,9090909091,
%T A355511 100000000000,1100000000000,12100000000000,133100000000000,
%U A355511 1464100000000000,16105100000000000,177156100000000000,1948717100000000000,21435888100000000000,235794769100000000000
%N A355511 a(n) is the number of monic polynomials of degree n over GF(11) without linear factors.
%F A355511 O.g.f. (1 - z)^(11)/(1-11*z) - 1
%e A355511 a(0) = 0 since there are no irreducible constant polynomials (as GF(11) is a field).
%e A355511 a(1) = 0 since all polynomials of degree 1 have linear factors.
%e A355511 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.
%t A355511 necklacePolynomial[q_, n_] :=
%t A355511   necklacePolynomial[q, n] = (1/n)*
%t A355511     DivisorSum[n, MoebiusMu[n/#1]*q^#1 & ];
%t A355511 numIrreds[q_, n_] := If[n != 0, necklacePolynomial[q, n], 0];
%t A355511 restrictedPolynomialsOGF[q_, n_, d_] :=
%t A355511   Product[(1 - z^If[ArrayDepth[d[[l]]] == 0, d[[l]], d[[l]][[1]]])^
%t A355511      If[ArrayDepth[d[[l]]] == 0, numIrreds[q, d[[l]]],
%t A355511              d[[l]][[2]]], {l, 1, Length[d]}]/(1 - q*z);
%t A355511 numRestrictedPolys[q_, n_, d_] :=
%t A355511   SeriesCoefficient[restrictedPolynomialsOGF[q, n, d], {z, 0, n}];
%t A355511 q = 11;
%t A355511 TableForm[{#, numRestrictedPolys[q, #, {1}]} & /@ (Range[20]),
%t A355511 TableHeadings -> {{Row[{"(q=", q, ")"}]}, {"n", "#rootless monics"}}]
%Y A355511 Cf. A355510.
%K A355511 nonn
%O A355511 0,3
%A A355511 _Greyson C. Wesley_, Jul 04 2022