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A037240 Molien series for 3-D group X1.

%I A037240 #54 Apr 13 2022 01:15:15
%S A037240 1,1,5,10,24,42,83,132,222,335,511,728,1047,1428,1956,2586,3414,4389,
%T A037240 5638,7084,8888,10966,13494,16380,19841,23751,28371,33566,39616,46376,
%U A037240 54177,62832,72726,83661,96045,109668,124999,141778,160538,181006,203742,228459,255788,285384
%N A037240 Molien series for 3-D group X1.
%C A037240 Also multidigraphs with 3 nodes and n arcs. - _Vladeta Jovovic_, Dec 27 1999
%C A037240 Also preference profiles with 3 alternatives and n agents (IANC model). - _Alexander Karpov_, Nov 23 2017
%H A037240 Robert Israel, <a href="/A037240/b037240.txt">Table of n, a(n) for n = 0..10000</a>
%H A037240 Ö. Egecioglu, <a href="https://www.cs.ucsb.edu/research/tech-reports/2005-25">Uniform generation of anonymous and neutral preference profiles for social choice rules</a>, Technical Report, UCSB, 2005.
%H A037240 Ö. Egecioglu, <a href="https://doi.org/10.1515/MCMA.2009.014">Uniform generation of anonymous and neutral preference profiles for social choice rules</a>, Monte Carlo Methods and Applications, 15(3), Jan 2009, 241-255.
%H A037240 Ira Gessel, <a href="https://mathoverflow.net/q/167518">Combinatorial counting with symmetry</a>, MathOverflow, 2014.
%H A037240 Marko V. Jaric and Joseph L. Birman, <a href="https://doi.org/10.1063/1.523443">Calculation of the Molien generating function for invariants of space groups</a>, J. Math. Phys. 18 (1977), 1459-1465; 2085.
%H A037240 Alexander V. Karpov, <a href="https://wp.hse.ru/data/2018/04/04/1164595187/188EC2018.pdf">An Informational Basis for Voting Rules</a>, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
%H A037240 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F A037240 G.f.: (1 + x^2 + 3*x^3 + 5*x^4 + x^5 + x^6)/((1 - x)*(1 - x^2)^3*(1 - x^3)^2).
%F A037240 From _Alexander Karpov_, Nov 18 2017: (Start)
%F A037240 if n ==  0  mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16 + (n+3)/9;
%F A037240 if n ==  3  mod 6, a(n) = C(n+5,5)/6 + (n+3)/9;
%F A037240 if n == 2,4 mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16;
%F A037240 if n == 1,5 mod 6, a(n) = C(n+5,5)/6.
%F A037240 (End)
%p A037240 S:= series((1+x^2+3*x^3+5*x^4+x^5+ x^6)/(1 - x)/(1 - x^2)^3/(1 - x^3)^2, x, 101):
%p A037240 seq(coeff(S,x,n),n=0..100); # _Robert Israel_, Nov 22 2017
%t A037240 CoefficientList[Series[(1 +x^2 +3x^3 +5x^4 +x^5 +x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2, {x, 0, 43}], x] (* _Michael De Vlieger_, Nov 01 2017 *)
%o A037240 (PARI) Vec((1+x^2+3*x^3+5*x^4+x^5+x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2 + O(x^50)) \\ _Michel Marcus_, Oct 31 2017
%o A037240 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 +x^2 +3*x^3 +5*x^4 +x^5 +x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) )); // _G. C. Greubel_, Jan 31 2020
%o A037240 (Sage)
%o A037240 def A037240_list(prec):
%o A037240     P.<x> = PowerSeriesRing(ZZ, prec)
%o A037240     return P( (1+x^2+3*x^3+5*x^4+x^5+x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) ).list()
%o A037240 A037240_list(50) # _G. C. Greubel_, Jan 31 2020
%Y A037240 Column k=3 of A333361.
%K A037240 nonn
%O A037240 0,3
%A A037240 _N. J. A. Sloane_
%E A037240 Terms a(35) and beyond from _Alexander Karpov_, Oct 29 2017