A232162 Number of Weyl group elements, not containing an s_r 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 B and rank n.

0, 0, 2, 3, 5, 14, 30, 62, 139, 305, 660, 1444, 3158, 6887, 15037, 32842, 71698, 156538, 341799, 746273, 1629384, 3557592, 7767594, 16959611, 37029365, 80849350, 176525142, 385422198, 841524755, 1837371729, 4011688220, 8759056412, 19124384574, 41755877375, 91169119405

Offset

0,3

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 [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.

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) = A232162(n-1) + A232162(n-2) + 3*A232162(n-3) + A232162(n-4).

From Colin Barker, Dec 31 2013: (Start)

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

G.f.: -x^2*(x + 2)/(x^4 + 3*x^3 + x^2 + x - 1). (End)

Example

For n=4, a(4) = A232162(3) + A232162(2) + 3*A232162(1) + A232162(0) = 3+2+3*0+0=5.

Maple

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

Mathematica

LinearRecurrence[{1, 1, 3, 1}, {0, 0, 2, 3}, 32] (* Jean-François Alcover, Nov 24 2017 *)

Prog

(PARI) Vec(-x^2*(x+2)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013

Crossrefs

Cf. A232163, A232164, A232165.

Sequence in context: A143743 A336273 A104870 * A243555 A114411 A323768

Adjacent sequences: A232159 A232160 A232161 * A232163 A232164 A232165

Keyword

nonn,easy

Author

Pamela E Harris, Nov 19 2013

Status

approved