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A008627 Molien series for A_4.

%I A008627 #29 Jul 08 2025 01:18:58
%S A008627 1,1,2,3,5,6,10,12,17,21,28,33,43,50,62,72,87,99,118,133,155,174,200,
%T A008627 222,253,279,314,345,385,420,466,506,557,603,660,711,775,832,902,966,
%U A008627 1043,1113,1198,1275,1367,1452,1552,1644,1753,1853,1970,2079,2205,2322
%N A008627 Molien series for A_4.
%C A008627 With offset = 4: a(n) is the number of equivalence classes of compositions (summands >=1) of n into exactly 4 parts where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions.  For example, let the class representatives be the last such composition in lexicographic order.  a(10)=10 because we have the following nine partitions of 10 into 4 parts, {7,1,1,1}, {6,2,1,1}, {5,3,1,1}, {5,2,2,1}, {4,4,1,1}, {4,3,2,1}, {4,2,2,2},{3,3,3,1}, {3,3,2,2} and the class represented by {3,4,2,1}. - _Geoffrey Critzer_, Oct 16 2012
%D A008627 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008627 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008627 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,-1,2,1,-1).
%F A008627 a(n) ~ 1/72*n^3. - _Ralf Stephan_, Apr 29 2014
%F A008627 G.f.: ( 1-x^2+x^4 ) / ( (1+x+x^2)*(1+x)^2*(x-1)^4 ). - _R. J. Mathar_, Dec 18 2014
%p A008627 (1+x^6)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4): seq(coeff(series(%,x,n+1),x,n), n=0..60);
%t A008627 nn=50;CoefficientList[Series[CycleIndex[AlternatingGroup[4],s]/.Table[s[i]->x^i/(1-x^i),{i,1,nn}],{x,0,nn}],x]  (* _Geoffrey Critzer_, Oct 16 2012 *)
%o A008627 (Sage)
%o A008627 ring = PowerSeriesRing(ZZ, 'x', default_prec=50)
%o A008627 ms = AlternatingGroup(4).molien_series()
%o A008627 list(ring(ms))
%o A008627 # _Ralf Stephan_, Apr 29 2014
%K A008627 nonn,easy
%O A008627 0,3
%A A008627 _N. J. A. Sloane_