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%I #36 Oct 29 2022 11:12:42
%S 0,0,21,112,819,5712,39991,279936,1959552,13716864,96018048,672126336,
%T 4704884352,32934190464,230539333248,1613775332736,11296427329152,
%U 79074991304064,553524939128448,3874674573899136,27122722017293952
%N a(n) is the number of monic polynomials of degree n over GF(7) without linear factors.
%H A. Knopfmacher and J. Knopfmacher, <a href="https://doi.org/10.1016/0012-365X(93)90227-K">Counting irreducible factors of polynomials over a finite field</a>, Discrete Mathematics, Volume 112, Issues 1-3, 1993, Pages 103-118.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (7).
%F O.g.f.: (1 - z)^7/(1 - 7*z) - 1.
%F For n >= 7, a(n) = 6^7 * 7^(n-7).
%e a(0) = 0 since all constant polynomials are units (as GF(7) is a field).
%e a(1) = 0 since all polynomials of degree 1 have linear factors.
%e a(2), the number of quadratic polynomials without linear factors, then coincides with the number of irreducible quadratics in GF(7), which is known to be M(7,2), where M(a,d) is the necklace polynomial, so a(2) = 21.
%t CoefficientList[Series[(1-z)^7/(1-7 z)-1,{z,0,15}]//Normal,z] (* For all terms *)
%t (7^(#-7)) &/@ Range[7,15]*6^7 (* For n>=7 *)
%t Join[{0,0,21,112,819,5712,39991},NestList[7#&,279936,20]] (* _Harvey P. Dale_, Oct 29 2022 *)
%K nonn,easy
%O 0,3
%A _Greyson C. Wesley_, Jul 04 2022
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