A364643 - OEIS

%I #29 Mar 25 2024 10:27:14

%S 1,2,4,22,102,474,2250,10910,53886,270322,1373970,7061542,36639702,

%T 191677386,1009942362,5354887470,28550730222,152979375842,

%U 823329316386,4448856020534,24126427982406,131270003806906,716377546590186,3920251765198782,21507301494123102,118269635529457874

%N Number of separable elements of the Weyl group of type D_n.

%H Christian Gaetz and Yibo Gao, <a href="https://arxiv.org/abs/1905.09331">Separable elements in Weyl groups</a>, arXiv:1905.09331 [math.CO], 2019.

%F 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).

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

%F a(n) = 2*A006318(n) - 3*A006318(n-1) - 2*A006318(n-2) (n >= 3) - _Fern Gossow_, Dec 05 2023

%F 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

%t b[0] = 1 ; a[0] = 1; a[1] = 2; a[2] = 4;

%t b[n_Integer] :=

%t  b[n] = b[n - 1] + Sum[b[k]*b[n - 1 - k], {k, 0, n - 1}];

%t a[n_Integer] :=

%t  a[n] = a[n - 1] + b[n - 2] + 2*b[n - 1] +

%t Sum[b[k]*a[n - k - 1], {k, 0, n - 3}]; Table[a[n], {n, 0, 30}] (* _Andy Huchala_, Aug 01 2023 *)

%o (Sage)

%o prec = 10

%o R.<x> = PowerSeriesRing(QQ,prec)

%o f = (3 - 4*x - 5*x^2 - (1 - x)*sqrt(1 - 6*x + x^2))/(1 - x + sqrt(1 - 6*x + x^2))

%o list(f) # _Andy Huchala_, Jul 31 2023

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

%K nonn

%O 0,2

%A _Fern Gossow_, Jul 30 2023