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

0,3

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.

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).

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)

Example

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

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 *)

Prog

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

Crossrefs

Cf. A232164.

Sequence in context: A091765 A368882 A036746 * A129955 A034066 A034076

Adjacent sequences: A232162 A232163 A232164 * A232166 A232167 A232168

Keyword

nonn,easy

Author

Pamela E Harris, Nov 19 2013

Status

approved