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%I #38 Sep 08 2022 08:44:34
%S 1,0,1,0,2,1,3,2,5,4,7,7,11,11,15,16,21,22,28,30,37,39,47,50,60,63,74,
%T 78,91,95,109,115,131,137,154,162,181,190,210,221,243,255,278,292,318,
%U 333,360,377,407,425,457,477,512,533,570,593,633,658,700,727
%N Molien series for a certain group of order 52.
%C The group is a semidirect product C13: C4 presented by <g, h | g^13=1, h^4=1, hg = g^5 h>. The group has 3 irreducible characters of degree 4, all of which have the same Molien series, this sequence. - _Eric M. Schmidt_, Feb 02 2013
%H Eric M. Schmidt, <a href="/A005916/b005916.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1,0,0,0,0,0,1,-1,-1,1,-1,1,1,-1).
%F G.f.: (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)). - _Colin Barker_, Jan 31 2013, confirmed and simplified by _Eric M. Schmidt_, Feb 02 2013
%F a(n) ~ (1/312)*n^3. - _Ralf Stephan_, Apr 29 2014
%p m:=60; S:=series((1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 06 2020
%t CoefficientList[Series[(1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)), {x, 0, 60}], x] (* _G. C. Greubel_, Feb 06 2020 *)
%t LinearRecurrence[{1,1,-1,1,-1,-1,1,0,0,0,0,0,1,-1,-1,1,-1,1,1,-1},{1,0,1,0,2,1,3,2,5,4,7,7,11,11,15,16,21,22,28,30},60] (* _Harvey P. Dale_, May 11 2022 *)
%o (GAP) series:=MolienSeries(First(Irr(SmallGroup(52,3)), irr->Degree(irr)=4));; List([0..30], i->ValueMolienSeries(series, i)); # _Eric M. Schmidt_, Feb 02 2013
%o (PARI) Vec( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) +O('x^60) ) \\ _G. C. Greubel_, Feb 06 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) )); // _G. C. Greubel_, Feb 06 2020
%o (Sage)
%o def A005916_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) ).list()
%o A005916_list(60) # _G. C. Greubel_, Feb 06 2020
%K nonn
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Eric M. Schmidt_, Feb 02 2013
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