A008668 - OEIS

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

%S 1,1,1,1,1,1,2,2,2,2,3,3,4,4,4,5,6,6,7,7,8,9,10,10,11,12,13,14,15,15,

%T 18,19,20,21,22,23,26,27,28,29,32,33,36,37,38,41,44,45,48,49,52,55,58,

%U 59,62,65,68,71,74,75,81,84,87,90,93,96,102,105,108,111,117,120,126,129,132,138

%N Molien series for 4-dimensional reflection group [3,3,5] of order 14400.

%C The relevant generating function is 1/((1-z^2)*(1-z^12)*(1-z^20)*(1-z^30)) and is reduced with x=z^2 below to indicate that the intermediate zeros are not listed.

%C Number of partitions into parts 1, 6, 10, and 15. - _Joerg Arndt_, Apr 29 2014

%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, no. 14. Springer Verlag, 1957, Table 10.

%D L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 30).

%H G. C. Greubel, <a href="/A008668/b008668.txt">Table of n, a(n) for n = 0..1000</a>

%H Roberto De Maria Nunes Mendes, <a href="https://doi.org/10.1090/S0002-9947-1975-0357687-1">Symmetries of spherical harmonics</a>, Transactions of the American Mathematical Society 204 (1975): 161-178. See subgroup 68.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=240">Encyclopedia of Combinatorial Structures 240</a>

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

%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1).

%F G.f.: 1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)). - _M. F. Hasler_, Mar 26 2012

%F a(n) ~ 1/5400*n^3. - _Ralf Stephan_, Apr 29 2014

%p seq(coeff(series(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)), x, n+1), x, n), n = 0 .. 80); # _G. C. Greubel_, Sep 08 2019

%t CoefficientList[Series[1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)), {x,0,80}], x] (* _G. C. Greubel_, Sep 08 2019 *)

%o (PARI) A008668_list = n -> Vec(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)) +O(x^n)) \\ _M. F. Hasler_, Mar 26 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)) )); // _G. C. Greubel_, Sep 08 2019

%o (Sage)

%o def A008668_list(prec):

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

%o     return P(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15))).list()

%o A008668_list(80) # _G. C. Greubel_, Sep 08 2019

%K nonn

%O 0,7

%A _N. J. A. Sloane_

%E Terms a(61) onward added by _G. C. Greubel_, Sep 08 2019