A232165 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra sp(2n).

0, 1, 2, 3, 8, 18, 37, 82, 181, 392, 856, 1873, 4086, 8919, 19480, 42530, 92853, 202742, 442665, 966496, 2110240, 4607473, 10059866, 21964555, 47957080, 104708706, 228619317, 499163818, 1089866333, 2379596808, 5195573912, 11343933537, 24768164206, 54078416287

Offset: 0

Author

Pamela E Harris, Nov 19 2013

Keywords

• nonn
• easy

Comments

Number of Weyl group elements contributing nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the adjoint representation for the Lie algebra of type C and rank n.

Examples

			For n=3, a(3) = A232164(3) + A232164(2) = 2+1 = 3.
		

References

• P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

Links

• Stefano Spezia, Table of n, a(n) for n = 0..2950
• 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, 2013
• B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
• László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 1 at p. 3.
• Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

Crossrefs

Cf. A232164.

Programs

Maple

r:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
elif n=2 then return 1:
elif n=3 then return 2:
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=0 then return 0:
elif n=1 then return 1:
else return
r(n)+r(n-1):
end if;
end proc:

Mathematica

LinearRecurrence[{1,1,3,1},{0,1,2,3},40] (* Harvey P. Dale, Nov 22 2014 *)

PARI

Vec(-x*(x+1)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Jan 01 2014

Formula

a(n) = A232164(n) + A232164(n-1).

From Colin Barker, Jan 01 2014: (Start)

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

G.f.: -x*(x + 1)/(x^4 + 3*x^3 + x^2 + x - 1). (End)