A316160 - OEIS

%I #27 Apr 19 2023 02:41:51

%S 6,26,80,226,600,1528,3776,9120,21632,50560,116736,266752,604160,

%T 1357824,3031040,6725632,14843904,32604160,71303168,155320320,

%U 337117184,729284608,1572864000,3382706176,7256145920,15527313408,33151778816,70632079360,150189637632,318767104000,675383607296

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

%C 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.

%C 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.

%H Vincent Coll, et al., <a href="https://doi.org/10.4172/1736-4337.1000227">Meander graphs and Frobenius seaweed Lie algebras II</a>, Journal of Generalized Lie Theory and Applications 9.1 (2015).

%H Vladimir Dergachev, and Alexandre Kirillov, <a href="https://www.emis.de/journals/JLT/vol.10_no.2/6.html">Index of Lie algebras of seaweed type</a>, J. Lie Theory 10.2 (2000): 331-343.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8)

%F 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.

%F G.f.: 2*x^3*(3-5x-2x^2+5x^3-2x^4)/(1-2*x)^3. [Corrected by _Georg Fischer_, May 23 2019]

%F 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

%t 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 *)

%o (Sage)

%o [(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)]

%K easy,nonn

%O 3,1

%A _Nick Mayers_, _Andrew W. Mayers_, Jun 25 2018