cp's OEIS Frontend

A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).

%I A165921 #35 Oct 28 2018 17:08:58
%S A165921 0,0,0,1,1,3,4,9,15,31,53,105,189,363,672,1285,2407,4599,8704,16641,
%T A165921 31713,60787,116390,223696,429975,828495,1597440,3085465,5964488,
%U A165921 11545611,22368256,43383477,84212475,163617801,318140816,619094385,1205595657,2349383715,4581280972,8939118925,17452532040,34093383807
%N A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).
%C A165921 The terms are denoted h_6 in the Michon/Ravache reference.
%D A165921 J. E. Iglesias, Enumeration of polytypes MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 178-194, Table 1.
%H A165921 J.-F. Michon, P. Ravache, <a href="http://dx.doi.org/10.1016/j.ffa.2010.01.004">On different families of invariant irreducible polynomials over F_2[X]</a>, Finite fields & Applications, 16 (2010) 163-174.
%F A165921 (see PARI code)
%F A165921 a(p) = (2^(p-1)-1)/3p = A096060(n) for all primes p = prime(n) >= 5, n >= 3: A165921 o A000040 = A096060 on the domain [3..oo) of that sequence. - _M. F. Hasler_, Sep 27 2018
%t A165921 L[n_, k_] := DivisorSum[GCD[n, k], MoebiusMu[#]*Binomial[n/#, k/#] &];
%t A165921 A165920[n_] := Sum[If[(n + k) ~Mod~ 3 == 1, L[n, k], 0], {k, 0, n}]/n;
%t A165921 A001037[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[#]*2^(n/#) &]/n];
%t A165921 A000048[n_] := DivisorSum[n, Mod[#, 2]*(MoebiusMu[#]*2^(n/#)) &]/(2*n);
%t A165921 A165921[n_] := Module[{an},
%t A165921   If[n <= 2, Return[0]];
%t A165921   an = A001037[n];
%t A165921   If[Mod[n, 2] == 0, an -= 3*A000048[n/2]];
%t A165921   If[Mod[n, 3] == 0, an -= 2*A165920[n/3]];
%t A165921   an /= 6;
%t A165921   Return[an]
%t A165921 ];
%t A165921 Table[A165921[n], {n, 2, 50}] (* _Jean-François Alcover_, Dec 02 2015, adapted from _Joerg Arndt_'s PARI script *)
%o A165921 (PARI)
%o A165921 L(n, k)=sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
%o A165921 A165920(n)=sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
%o A165921 A001037(n)=if(n<1, n==0, sumdiv(n, d, moebius(d)*2^(n/d))/n);
%o A165921 A000048(n)=sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n);
%o A165921 A165921(n)= /* this sequence */
%o A165921 {
%o A165921     my(an);
%o A165921     if ( n<=2, return(0) );
%o A165921     an = A001037(n);
%o A165921     if (n%2==0, an -= 3*A000048(n/2) );
%o A165921     if (n%3==0, an -= 2*A165920(n/3) );
%o A165921     an /= 6;
%o A165921     return( an );
%o A165921 }
%o A165921 /* _Joerg Arndt_, Jul 12 2012 */
%o A165921 (PARI) A165921(n)=if(n>2,A001037(n)-if(!bittest(n,0),3*A000048(n\2))-if(n%3==0,2*A165920(n\3)))\6 \\ Based on _Joerg Arndt_'s code from Jul 12 2012. Take up-to-date code for other sequences from the respective record. - _M. F. Hasler_, Sep 27 2018
%Y A165921 A001037 is the enumeration by degree of the irreducible polynomials over GF(2), A000048 is the number of 3-elements orbits, A165920 is the number of 2-elements orbits.
%Y A165921 Cf. A011957.
%Y A165921 Cf. A096060 = A165921 o A000040 (on 3..oo), a subsequence of this sequence.
%K A165921 easy,nonn
%O A165921 2,6
%A A165921 _Jean Francis Michon_, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009
%E A165921 Incorrect formula removed and more terms added by _Joerg Arndt_, Jul 12 2012