A038539 Complex semisimple Lie algebras of dimension n.

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, 21, 23, 25, 28, 34, 35, 37, 45, 45, 45, 56, 59, 61, 77, 80, 76, 92, 100, 101, 119, 130, 133, 147, 159, 170, 188, 200, 216, 243, 251, 260, 298, 317, 329, 379, 400

Offset

1,9

Comments

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

References

N. Jacobson, Lie Algebras, Dover Publications.

Links

Table of n, a(n) for n=1..67.

Formula

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)).

Mathematica

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. *)

Crossrefs

Sequence in context: A109697 A358903 A103373 * A275891 A109368 A046774

Adjacent sequences: A038536 A038537 A038538 * A038540 A038541 A038542

Keyword

nonn,easy,nice

Author

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

Status

approved