A234598 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra of so(2n).

9, 18, 35, 82, 180, 385, 846, 1853, 4034, 8810, 19249, 42014, 91727, 200298, 437316, 954809, 2084746, 4551801, 9938290, 21699138, 47377577, 103443386, 225856667, 493131922, 1076696324, 2350841633, 5132790390, 11206852917, 24468864530

Offset: 4

Author

Erik Insko, Dec 28 2013

Keywords

• nonn
• easy

Comments

Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra of type D and rank n.

Examples

			For n = 8, a(n) = 107+73 = 180 and a(n) = 3(34) + 2(14) + 6(7) + 2(4) = 180.
		

Links

• P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
• P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
• B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
• Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

Crossrefs

Cf. A232163, A232165, A234576, A234597.

Programs

Maple

r:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 4:
elif n=5 then return 7:
elif n=6 then return 14:
elif n=7 then return 34:
else return
r(n-1)+r(n-2)+3*r(n-3)+r(n-4):
end if;
end proc:
a:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 9:
elif n=5 then return 18:
elif n=6 then return 35:
elif n=5 then return 82:
else return
3*r(n-1)+2*r(n-2)+6*r(n-3)+2*r(n-4):
end if;
end proc:

Mathematica

LinearRecurrence[{1, 1, 3, 1}, {9, 18, 35, 82}, 30] (* Jean-François Alcover, Dec 06 2017 *)

Formula

a(n) = A234597(n) + A234576(n).

a(n) = 3*A234576(n-1) + 2*A234576(n-2) + 6*A234576(n-3) + 2*A234576(n-4).

G.f.: x^4*(2*x^3 + 8*x^2 + 9*x + 9)/(-x^4 - 3*x^3 - x^2 - x + 1). - Ralf Stephan, Jan 05 2014

Extensions

More terms from Ralf Stephan, Jan 05 2014