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A008669 Molien series for 4-dimensional complex reflection group of order 7680 (in powers of x^4). 3

%I

%S 1,1,2,3,4,6,8,10,13,16,20,24,29,34,40,47,54,62,71,80,91,102,114,127,

%T 141,156,172,189,207,226,247,268,291,315,340,367,395,424,455,487,521,

%U 556,593,631,671,713,756,801,848,896,947,999,1053,1109,1167,1227,1289

%N Molien series for 4-dimensional complex reflection group of order 7680 (in powers of x^4).

%C Number of partitions of n into parts 1, 2, 3 and 5. - _David Neil McGrath_, Sep 15 2014

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,3,5).

%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 29).

%H Vincenzo Librandi, <a href="/A008669/b008669.txt">Table of n, a(n) for n = 0..10000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=239">Encyclopedia of Combinatorial Structures 239</a>

%H Atsuto Seko, Atsushi Togo, Isao Tanaka, <a href="https://arxiv.org/abs/1901.02118">Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential</a>, arXiv:1901.02118 [physics.comp-ph], 2019.

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,0,0,-1,0,1,1,-1).

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

%F a(n) = round((n+3)*(2*n+9)*(n+9)/360).

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)).

%F a(n) = -a(-11-n).

%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-7) + a(n-9) + a(n-10) - a(n-11). - _David Neil McGrath_, Sep 15 2014

%e There are 6 partitions of 5 into parts 1,2,3 and 5. These are (5)(32)(311)(221)(2111)(11111). - _David Neil McGrath_, Sep 15 2014

%p 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)); seq(coeff(series(%, x, n+1), x, n), n = 0..60); # modified by _G. C. Greubel_, Sep 08 2019

%t LinearRecurrence[{1,1,0,-1,0,0,-1,0,1,1,-1}, {1,1,2,3,4,6,8,10,13,16, 20}, 60] (* _Harvey P. Dale_, Feb 25 2015 *)

%o (PARI) a(n)=round((n+3)*(2*n+9)*(n+9)/360)

%o (MAGMA) [Round((n+3)*(2*n+9)*(n+9)/360): n in [0..60]]; // _Vincenzo Librandi_, Jun 23 2011

%o (Sage) [round((n+3)*(2*n+9)*(n+9)/360) for n in (0..60)] # _G. C. Greubel_, Sep 08 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified May 31 20:23 EDT 2020. Contains 334748 sequences. (Running on oeis4.)