A234597 Number of Weyl group elements, not containing an s_1 factor, 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.

5, 11, 21, 48, 107, 229, 501, 1099, 2394, 5225, 11417, 24923, 54409, 118808, 259403, 566361, 1236597, 2699975, 5895058, 12871185, 28102765, 61359099, 133970477, 292509056, 638659595, 1394439181, 3044596421, 6647523443, 14514097002, 31689848889, 69191112641

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, 2013

B. Kostant, A Formula for the Multiplicity of a Weight, Proc. Natl. Acad. Sci. USA, 44 (No. 6, June 1958), 588-589.

Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

Formula

a(n) = A234576(n) + A234576(n-1).

a(n) = a(n-1)+a(n-2)+3*a(n-3)+a(n-4). G.f.: -x^4*(x^3+5*x^2+6*x+5) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Dec 30 2013

Example

For n=6, a(6) = A234576(6) + A234576(5)= 14+7 = 21.

Maple

r:=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
r(n-1)+r(n-2)+3*r(n-3)+r(n-4):
end if;
end proc:
a:=proc(n::nonnegint)
if n<=3 then return 0:
elif n=4 then return 5:
elif n=5 then return 11:
else return
r(n)+r(n-1):
end if;
end proc:

Mathematica

LinearRecurrence[{1, 1, 3, 1}, {5, 11, 21, 48}, 40] (* Harvey P. Dale, Feb 17 2016 *)

Prog

(PARI) Vec(-x^4*(x^3+5*x^2+6*x+5)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 30 2013

Crossrefs

Cf. A234576.

Sequence in context: A131898 A168642 A357750 * A261982 A296033 A296968

Adjacent sequences: A234594 A234595 A234596 * A234598 A234599 A234600

Keyword

nonn,easy

Author

Erik Insko, Dec 28 2013

Status

approved