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Complex semisimple Lie algebras of dimension n.
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%I #9 Jun 26 2013 11:17:18

%S 1,0,1,1,0,1,1,1,2,2,2,2,2,3,4,4,4,5,5,5,8,8,7,12,12,9,13,15,15,19,21,

%T 21,23,25,28,34,35,37,45,45,45,56,59,61,77,80,76,92,100,101,119,130,

%U 133,147,159,170,188,200,216,243,251,260,298,317,329,379,400

%N Complex semisimple Lie algebras of dimension n.

%C Direct consequence of classification of complex finite-dimensional simple Lie algebras.

%D N. Jacobson, Lie Algebras, Dover Publications.

%F G.f.: (1+x)/((1 - x^14)(1 - x^52)(1 - x^78)(1-x^133)(1 - x^248) prod( 1-x^(n^2 + 2n), n = 1..inf) prod(1 - x^(2n^2 + n), n=2..inf) prod(1-x^(2n^2+n), n=3..inf) prod( 1-x^(2n^2 - n), n=4..inf)).

%t max = 67; dims = {14, 52, 78, 133, 248}; f[x_] := (1 + x)/(Times @@ (1 - x^dims)*Product[1 - x^(n^2 + 2*n), {n, 1, max}]*Product[1 - x^(2*n^2 + n), {n, 2, max}]*Product[1 - x^(2*n^2 + n), {n, 3, max}]*Product[1 - x^(2*n^2 - n), {n, 4, max}]); CoefficientList[ Series[f[x], {x, 0, max}], x] // Rest (* _Jean-François Alcover_, Oct 08 2012, from g.f. *)

%K nonn,easy,nice

%O 1,9

%A Paolo Dominici (pl.dm(AT)libero.it)