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A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). 14

%I #50 Sep 08 2022 08:44:36

%S 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 53,60,63,71,74,83,87,96,101,111,116,127,133,145,151,164,171,185,193,

%U 207,216,232,241,258,268,286,297,316,328,348,361,382,396,419,433,457

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

%C 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 Number of partitions into parts 2, 3, 4, and 5. - _Joerg Arndt_, Apr 29 2014

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

%H Harvey P. Dale, <a href="/A008667/b008667.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=241">Encyclopedia of Combinatorial Structures 241</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <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 Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - _Michael Somos_, Sep 23 2006

%F a(-14 - n) = -a(n). - _Michael Somos_, Sep 23 2006

%F a(n) ~ 1/720*n^3. - _Ralf Stephan_, Apr 29 2014

%F 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 a(n)-a(n-2) = A008680(n). - _R. J. Mathar_, Jun 23 2021

%F a(n)-a(n-3) = A025802(n). - _R. J. Mathar_, Jun 23 2021

%F a(n)-a(n-4) = A025795(n). - _R. J. Mathar_, Jun 23 2021

%F a(n)-a(n-5) = A005044(n+3). - _R. J. Mathar_, Jun 23 2021

%e 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 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 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 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 CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* _Harvey P. Dale_, Feb 22 2018 *)

%o (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 (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 (Sage)

%o def A008667_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()

%o A008667_list(65) # _G. C. Greubel_, Sep 08 2019

%Y Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

%Y Cf. A005044, A001401 (partial sums).

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)