

A355510


a(n) is the number of monic polynomials of degree n over GF(7) without linear factors.


1



0, 0, 21, 112, 819, 5712, 39991, 279936, 1959552, 13716864, 96018048, 672126336, 4704884352, 32934190464, 230539333248, 1613775332736, 11296427329152, 79074991304064, 553524939128448, 3874674573899136, 27122722017293952
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OFFSET

0,3


LINKS



FORMULA

O.g.f.: (1  z)^7/(1  7*z)  1.
For n >= 7, a(n) = 6^7 * 7^(n7).


EXAMPLE

a(0) = 0 since all constant polynomials are units (as GF(7) 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(7), which is known to be M(7,2), where M(a,d) is the necklace polynomial, so a(2) = 21.


MATHEMATICA

CoefficientList[Series[(1z)^7/(17 z)1, {z, 0, 15}]//Normal, z] (* For all terms *)
(7^(#7)) &/@ Range[7, 15]*6^7 (* For n>=7 *)
Join[{0, 0, 21, 112, 819, 5712, 39991}, NestList[7#&, 279936, 20]] (* Harvey P. Dale, Oct 29 2022 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



