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A008720 Molien series for 3-dimensional group [2,5] = *225.

%I A008720 #24 Jun 20 2025 10:50:08
%S A008720 1,0,2,0,3,1,4,2,5,3,7,4,9,5,11,7,13,9,15,11,18,13,21,15,24,18,27,21,
%T A008720 30,24,34,27,38,30,42,34,46,38,50,42,55,46,60,50,65,55,70,60,75,65,81,
%U A008720 70,87,75,93,81,99,87,105,93,112,99,119,105,126,112,133,119,140,126,148,133,156
%N A008720 Molien series for 3-dimensional group [2,5] = *225.
%C A008720 a(n) is the number of partitions of n into parts 2 and 5 where there are two kinds of parts 2. - _Hoang Xuan Thanh_, Jun 20 2025
%H A008720 G. C. Greubel, <a href="/A008720/b008720.txt">Table of n, a(n) for n = 0..1000</a>
%H A008720 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=223">Encyclopedia of Combinatorial Structures 223</a>
%H A008720 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008720 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1,1,0,-2,0,1).
%F A008720 a(n) = floor((n^2 + n*(9+5*(-1)^n) + 23*(-1)^n + 26)/40). - _Hoang Xuan Thanh_, Jun 20 2025
%p A008720 1/((1-x^2)^2*(1-x^5)); seq(coeff(series(%, x, n+1), x, n), n = 0 .. 80); # modified by _G. C. Greubel_, Sep 09 2019
%t A008720 LinearRecurrence[{0,2,0,-1,1,0,-2,0,1}, {1,0,2,0,3,1,4,2,5}, 80] (* _Harvey P. Dale_, Dec 10 2015 *)
%o A008720 (PARI) my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^5))) \\ _G. C. Greubel_, Sep 09 2019
%o A008720 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^5)) )); // _G. C. Greubel_, Sep 09 2019
%o A008720 (Sage)
%o A008720 def A008720_list(prec):
%o A008720     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008720     return P(1/((1-x^2)^2*(1-x^5))).list()
%o A008720 A008720_list(80) # _G. C. Greubel_, Sep 09 2019
%o A008720 (GAP) a:=[1,0,2,0,3,1,4,2,5];; for n in [10..80] do a[n]:=2*a[n-2]-a[n-4] +a[n-5]-2*a[n-7]+a[n-9]; od; a; # _G. C. Greubel_, Sep 09 2019
%K A008720 nonn,easy
%O A008720 0,3
%A A008720 _N. J. A. Sloane_
%E A008720 Terms a(65) onward added by _G. C. Greubel_, Sep 09 2019