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A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

%I A008667 #57 Aug 21 2025 10:01:44
%S A008667 1,0,1,1,2,2,3,3,5,5,7,7,10,10,13,14,17,18,22,23,28,29,34,36,42,44,50,
%T A008667 53,60,63,71,74,83,87,96,101,111,116,127,133,145,151,164,171,185,193,
%U A008667 207,216,232,241,258,268,286,297,316,328,348,361,382,396,419,433,457
%N A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
%C A008667 Also, Molien series for invariants of finite Coxeter group A_4. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k. - _N. J. A. Sloane_, Jan 11 2016
%C A008667 Number of partitions into parts 2, 3, 4, and 5. - _Joerg Arndt_, Apr 29 2014
%D A008667 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
%D A008667 L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).
%H A008667 Harvey P. Dale, <a href="/A008667/b008667.txt">Table of n, a(n) for n = 0..1000</a>
%H A008667 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=241">Encyclopedia of Combinatorial Structures 241</a>
%H A008667 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008667 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,0,-1,-2,-1,0,1,1,1,0,-1).
%F A008667 Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - _Michael Somos_, Sep 23 2006
%F A008667 a(-14 - n) = -a(n). - _Michael Somos_, Sep 23 2006
%F A008667 a(n) ~ 1/720*n^3. - _Ralf Stephan_, Apr 29 2014
%F A008667 a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - _David Neil McGrath_, Sep 13 2014
%F A008667 From _R. J. Mathar_, Jun 23 2021: (Start)
%F A008667 a(n)-a(n-2) = A008680(n).
%F A008667 a(n)-a(n-3) = A025802(n).
%F A008667 a(n)-a(n-4) = A025795(n).
%F A008667 a(n)-a(n-5) = A005044(n+3). (End)
%F A008667 a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - _Hoang Xuan Thanh_, Aug 20 2025
%e A008667 a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
%e A008667 G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .
%p A008667 seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # _G. C. Greubel_, Sep 08 2019
%t A008667 SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
%t A008667 CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* _Harvey P. Dale_, Feb 22 2018 *)
%o A008667 (PARI) {a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* _Michael Somos_, Oct 14 2006 */
%o A008667 (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // _G. C. Greubel_, Sep 08 2019
%o A008667 (Sage)
%o A008667 def A008667_list(prec):
%o A008667     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008667     return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
%o A008667 A008667_list(65) # _G. C. Greubel_, Sep 08 2019
%Y A008667 Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.
%Y A008667 Cf. A005044, A001401 (partial sums).
%K A008667 nonn,easy,changed
%O A008667 0,5
%A A008667 _N. J. A. Sloane_