A232163 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra so(2n+1).

0, 1, 2, 5, 10, 22, 49, 106, 231, 506, 1104, 2409, 5262, 11489, 25082, 54766, 119577, 261078, 570035, 1244610, 2717456, 5933249, 12954570, 28284797, 61756570, 134838326, 294403857, 642796690, 1403472095, 3064318682, 6690584704

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 B and rank n.

Links

Table of n, a(n) for n=0..30.

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).

Formula

a(n) = A232162(n) + A232162(n-1) + A232162(n-2).

a(n) = a(n-1)+a(n-2)+3*a(n-3)+a(n-4). G.f.: -x*(2*x^2+x+1) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Jan 01 2014

Example

For n=8, a(8) = A232162(8) + A232162(7) + A232162(6) = 139+62+30 = 231.

Maple

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

Mathematica

LinearRecurrence[{1, 1, 3, 1}, {0, 1, 2, 5}, 31] (* Jean-François Alcover, Nov 26 2017 *)

Prog

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

Crossrefs

Cf. A232162.

Sequence in context: A341020 A123491 A329185 * A166300 A038149 A046745

Adjacent sequences: A232160 A232161 A232162 * A232164 A232165 A232166

Keyword

nonn,easy

Author

Pamela E Harris, Nov 19 2013

Status

approved