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A008643 Molien series for group of 4 X 4 upper triangular matrices over GF(2).

%I A008643 #58 Aug 15 2025 15:58:35
%S A008643 1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,35,35,44,44,56,56,68,68,84,
%T A008643 84,100,100,120,120,140,140,165,165,190,190,220,220,250,250,286,286,
%U A008643 322,322,364,364,406,406,455,455,504,504,560,560,616,616,680,680,744
%N A008643 Molien series for group of 4 X 4 upper triangular matrices over GF(2).
%C A008643 Number of partitions of n into parts 1, 2, 4 and 8. - _Ilya Gutkovskiy_, May 24 2017
%D A008643 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008643 Alois P. Heinz, <a href="/A008643/b008643.txt">Table of n, a(n) for n = 0..1000</a>
%H A008643 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=233">Encyclopedia of Combinatorial Structures 233</a>
%H A008643 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008643 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1).
%F A008643 G.f.: 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)).
%F A008643 a(n) = floor(((n+14)*(3*(n+1)*(-1)^n + 2*n^2 + 17*n + 57) + 24*(floor(n/2) + 1)*(-1)^floor(n/2))/768). - _Tani Akinari_, Jun 16 2013
%F A008643 a(n) ~ 1/384*n^3. - _Ralf Stephan_, Apr 29 2014
%p A008643 a:= proc(n) local m, r; m := iquo(n, 8, 'r'); r:= iquo(r,2)+1; ([11, 17, 26, 35][r]+ (9+ 3*r+ 4*m) *m) *m/3+ [1, 2, 4, 6][r] end: seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 06 2008
%t A008643 CoefficientList[1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)) + O[x]^65, x] (* _Jean-François Alcover_, May 29 2015 *)
%t A008643 LinearRecurrence[{1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1}, {1,1,2,2,4,4,6,6,10,10,14,14,20,20,26}, 65] (* _Ray Chandler_, Jul 15 2015 *)
%o A008643 (PARI) my(x='x+O('x^65)); Vec(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8))) \\ _G. C. Greubel_, Feb 01 2020
%o A008643 (PARI) my(b(m) = (m^3 + 12*m^2 + (44 - 3*(m%2))*m + 48)\48); vector(59,n,b((n-1)\2)) \\ _Hoang Xuan Thanh_, Aug 14 2025
%o A008643 (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (&*[1/(1-x^(2^j)): j in [0..3]]) )); // _G. C. Greubel_, Feb 01 2020
%o A008643 (Sage)
%o A008643 def A077952_list(prec):
%o A008643     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008643     return P( 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)) ).list()
%o A008643 A077952_list(65) # _G. C. Greubel_, Feb 01 2020
%Y A008643 Cf. A008804, A088932 (partial sums).
%K A008643 nonn,easy
%O A008643 0,3
%A A008643 _N. J. A. Sloane_