A316160 Number of pairs of compositions of n corresponding to a seaweed algebra of index n-3.

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

Table of n, a(n) for n=3..33.

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.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8)

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

Adjacent sequences: A316157 A316158 A316159 * A316161 A316162 A316163

Keyword

easy,nonn

Author

Nick Mayers, Andrew W. Mayers, Jun 25 2018

Status

approved