A364643 Number of separable elements of the Weyl group of type D_n.

1, 2, 4, 22, 102, 474, 2250, 10910, 53886, 270322, 1373970, 7061542, 36639702, 191677386, 1009942362, 5354887470, 28550730222, 152979375842, 823329316386, 4448856020534, 24126427982406, 131270003806906, 716377546590186, 3920251765198782, 21507301494123102, 118269635529457874

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Christian Gaetz and Yibo Gao, Separable elements in Weyl groups, arXiv:1905.09331 [math.CO], 2019.

Formula

a(n) = a(n-1) + A006318(n-2) + 2*A006318(n-1) + Sum_{k=0..n-3} A006318(k)*a(n-k-1) (n >= 3).

G.f.: (3 - 4*x - 5*x^2 - (1 - x)*sqrt(1 - 6*x + x^2))/(1 - x + sqrt(1 - 6*x + x^2)).

a(n) = 2*A006318(n) - 3*A006318(n-1) - 2*A006318(n-2) (n >= 3) - Fern Gossow, Dec 05 2023

D-finite with recurrence +2*(n+1)*a(n) +3*(-5*n+3)*a(n-1) +(18*n-43)*a(n-2) +3*(3*n-14)*a(n-3) +2*(-n+6)*a(n-4)=0. - R. J. Mathar, Mar 25 2024

Mathematica

b[0] = 1 ; a[0] = 1; a[1] = 2; a[2] = 4;
b[n_Integer] :=
 b[n] = b[n - 1] + Sum[b[k]*b[n - 1 - k], {k, 0, n - 1}];
a[n_Integer] :=
 a[n] = a[n - 1] + b[n - 2] + 2*b[n - 1] +
Sum[b[k]*a[n - k - 1], {k, 0, n - 3}]; Table[a[n], {n, 0, 30}] (* Andy Huchala, Aug 01 2023 *)

Prog

(Sage)
prec = 10
R.<x> = PowerSeriesRing(QQ, prec)
f = (3 - 4*x - 5*x^2 - (1 - x)*sqrt(1 - 6*x + x^2))/(1 - x + sqrt(1 - 6*x + x^2))
list(f) # Andy Huchala, Jul 31 2023

Crossrefs

Types A_n and B_n/C_n are counted by A006318 (large Schröder numbers).

Sequence in context: A152104 A047035 A080042 * A324145 A366732 A165588

Adjacent sequences: A364640 A364641 A364642 * A364644 A364645 A364646

Keyword

nonn

Author

Fern Gossow, Jul 30 2023

Status

approved