A232162 -id:A232162 - cp's OEIS frontend

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

Pamela E Harris, Nov 19 2013

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.

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

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).

Cf. A232162.

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:

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

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

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

A243094 Cardinality of the Weyl alternation set corresponding to the zero-weight in the representation of the Lie algebra sp(2n) whose highest weight is the second fundamental weight.

Original entry on oeis.org

1, 2, 5, 8, 19, 44, 92, 201, 444, 965, 2104, 4602, 10045, 21924, 47879, 104540, 228236, 498337, 1088072, 2375657, 5186976, 11325186, 24727205, 53988976, 117878715, 257374492, 561947340, 1226946953, 2678896484, 5849059949, 12770744632, 27883440986, 60880261949

Offset: 0

Pamela E Harris, Aug 18 2014

nonn

Number of Weyl group elements contributing nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the defining representation for the Lie algebra of type C and rank n. Here the highest weight would be the second fundamental weight of sp(2n).

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

Cf. A232162.

r:=proc(n::nonnegint) option remember
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):
end if;
end proc:

Join[{1}, LinearRecurrence[{1, 1, 3, 1}, {2, 5, 8, 19}, 32]] (* Jean-François Alcover, Dec 05 2017 *)

Vec( (x^4+2*x^3-2*x^2-x-1) / (x^4+3*x^3+x^2+x-1) +O(x^66) ) \\ Joerg Arndt, Aug 18 2014

a(n) = A232162(n) + A232162(n-1).
a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: (x^4 + 2*x^3 - 2*x^2 - x - 1) / (x^4 + 3*x^3 + x^2 + x - 1). - Joerg Arndt, Aug 18 2014

cp's OEIS Frontend

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

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