A027381 Number of irreducible polynomials of degree n over GF(9); dimensions of free Lie algebras.

1, 9, 36, 240, 1620, 11808, 88440, 683280, 5380020, 43046640, 348672528, 2852823600, 23535749880, 195528140640, 1634056262280, 13726075468992, 115813759112820, 981010688215680, 8338590828280440, 71097458824894320

Offset

0,2

Comments

Number of aperiodic necklaces with n beads of 9 colors. - Herbert Kociemba, Nov 25 2016

References

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1051 (terms 0..200 from T. D. Noe)

A. Pakapongpun, T. Ward, Functorial Orbit Counting, JIS 12 (2009) 09.2.4, example 3.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.

Index entries for sequences related to Lyndon words

Formula

G.f.: k=9, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

a(n) = Sum_{d|n} mu(d)*9^(n/d)/n for n > 0. - Andrew Howroyd, Oct 13 2017

Example

G.f. = 1 + 9*x + 36*x^2 + 240*x^3 + 1620*x^4 + 11808*x^5 + 88440*x^6 + ...

Mathematica

f[n_] := (1/n)*Sum[ MoebiusMu[d]*9^(n/d), {d, Divisors[n]}]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Jul 28 2014 *)
mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 9], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *)

Prog

(PARI) a(n) = if(n, sumdiv(n, d, moebius(d)*9^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015

Crossrefs

Column 9 of A074650.

Cf. A001037.

Sequence in context: A038780 A073984 A036907 * A335783 A024120 A262782

Adjacent sequences: A027378 A027379 A027380 * A027382 A027383 A027384

Keyword

nonn

Author

N. J. A. Sloane

Status

approved