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%I #19 Oct 22 2022 15:00:55
%S 1,1,1,3,3,3,6,6,8,12,14,18,23,27,34,43,52,62,79,93,109,138,159,187,
%T 236,270,316,385,443,518,620,719,836,983,1138,1314,1541,1770,2041,
%U 2388,2726,3122,3628,4124,4720,5459,6204,7063,8116,9203,10440,11940,13525,15306,17436,19690,22231,25208,28388,32013,36217,40673,45729,51575,57808,64817
%N a(n) gives the number of nonisomorphic connected compact Lie groups of dimension n which are simple products.
%C By the structure theorem for compact Lie groups, every compact connected Lie group is a finite central quotient of a product of copies of the circle group U(1) and compact simple Lie groups which are all known due to Cartan's classification. This sequence counts only those which are direct products of such groups.
%F G.f.: 1/((1-x)*(1-x^3)^2*(1-x^8)^2*(1-x^10)^2*(1-x^14)*...) = (1/(1-x)) * Product_{k>=0} (1-x^k)^A178176(k) with (1-x^k)^0 taken to be 1.
%e For n=0, the trivial group is the only such group.
%e For n=8, the 8 Lie groups are U(1)^8, U(1)^5 x SU(2), U(1)^5 x SO(3), U(1)^2 x SU(2)^2, U(1)^2 x SU(2) x SO(3), U(1)^2 x SO(3)^2, SU(3) and SU(3)/3.
%Y See also A178176 for enumeration of the simple factors giving these counts.
%K nonn
%O 0,4
%A _Andrew Rupinski_, Dec 18 2010
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