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A316160
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-3.
0
6, 26, 80, 226, 600, 1528, 3776, 9120, 21632, 50560, 116736, 266752, 604160, 1357824, 3031040, 6725632, 14843904, 32604160, 71303168, 155320320, 337117184, 729284608, 1572864000, 3382706176, 7256145920, 15527313408, 33151778816, 70632079360, 150189637632, 318767104000, 675383607296
OFFSET
3,1
COMMENTS
The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)), where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([_,_]), where [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
LINKS
Vincent Coll, et al., Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9.1 (2015).
Vladimir Dergachev, and Alexandre Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10.2 (2000): 331-343.
FORMULA
a(n) = (7*n-15)*2^(n-3) for 3 <= n <= 5 and a(n) = ((1/2)*n^2+(11/4)*n-(25/4))*2^(n-3) for n >= 5.
G.f.: 2*x^3*(3-5x-2x^2+5x^3-2x^4)/(1-2*x)^3. [Corrected by Georg Fischer, May 23 2019]
E.g.f.: (75 + 72*x - 30*x^2 - 8*x^3 + 2*x^4 - 3*exp(2*x)*(25 - 26*x - 8*x^2))/96. - Stefano Spezia, Nov 16 2022
MATHEMATICA
CoefficientList[Series[2*x^3*(3-5x-2x^2+5x^3-2x^4)/(1-2*x)^3, {x, 0, 33}], x] (* Georg Fischer, May 23 2019 *)
PROG
(Sage)
[(7*n-15)*2^(n-3) if n < 5 else ((1/2)*n^2+(11/4)*n-(25/4))*2^(n-3) for n in range(3, 263)]
CROSSREFS
Sequence in context: A335648 A094162 A229572 * A224035 A172207 A060101
KEYWORD
easy,nonn
AUTHOR
STATUS
approved