A234576 Number of Weyl group elements, not containing s_1 or s_2, which contribute 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 D and rank n.

4, 7, 14, 34, 73, 156, 345, 754, 1640, 3585, 7832, 17091, 37318, 81490, 177913, 388448, 848149, 1851826, 4043232, 8827953, 19274812, 42084287, 91886190, 200622866, 438036729, 956402452, 2088193969, 4559329474, 9954767528, 21735081361, 47456031280

Offset

4,1

Links

Table of n, a(n) for n=4..34.

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

G.f.: -x^4*(x^3+3*x^2+3*x+4) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Dec 30 2013

Example

For n = 8, a(8) = 34+14+3*7+4 = 73.

Maple

a:=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
a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
end if;
end proc:

Mathematica

LinearRecurrence[{1, 1, 3, 1}, {4, 7, 14, 34}, 31] (* Jean-François Alcover, Nov 26 2017 *)

Prog

(PARI) Vec(-x^4*(x^3+3*x^2+3*x+4)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 30 2013

Crossrefs

Sequence in context: A245002 A199628 A049945 * A076586 A240266 A064961

Adjacent sequences: A234573 A234574 A234575 * A234577 A234578 A234579

Keyword

nonn,easy

Author

Erik Insko, Dec 28 2013

Status

approved