A137207 - OEIS

A137207

Number of exceptional sets of roots of type D_n. Also the number of unordered factorizations of the Coxeter element.

12, 87, 584, 3835, 25008, 162792, 1060048, 6910695, 45119100, 295038315, 1932260256, 12673336052, 83236707232, 547388545740, 3604063891104, 23755630474079, 156740823815940, 1035157282013085, 6842413166034600, 45265133475699795, 299671339559444160, 1985322768625822080

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OFFSET

3,1

LINKS

Table of n, a(n) for n=3..24.

F. Chapoton Cluster algebras, 2002.

FORMULA

a(n) = (2*(n-1)/(2*n-1))*binomial(3*n-3,n-1)-binomial(3*n-5,n-2)+4*binomial(3*n-3,n-3).

a(n) = (16*n^2-41*n+24)/(n*(2*n-1))*binomial(3*n-5,n-2).

a(n) ~ 3^(3*n-9/2) / (2^(2*n-5) * sqrt(Pi*n)). - Amiram Eldar, Oct 30 2025

EXAMPLE

a(3) = 12 because D3 is the same as A3.

MATHEMATICA

a[n_] := (16*n^2-41*n+24)/(n*(2*n-1)) * Binomial[3*n-5, n-2]; Array[a, 25, 3] (* Amiram Eldar, Oct 30 2025 *)

PROG

(MuPAD)

modu_NC_D:=proc(n) begin (16*n*n-41*n+24)/n/(2*n-1)*binomial(3*n-5, n-2) end;

(SageMath)

def A137207(n):

return (16*n*n-41*n+24)*binomial(3*n-5, n-2)/n/(2*n-1)

CROSSREFS

Cf. A001764 (type A), A045721 (type B).

Sequence in context: A383137 A183721 A180797 * A206765 A228500 A348415

Adjacent sequences: A137204 A137205 A137206 * A137208 A137209 A137210

KEYWORD

nonn,easy

AUTHOR

F. Chapoton, Mar 05 2008

EXTENSIONS

a(22)-a(24) from Stefano Spezia, Feb 29 2024

STATUS

approved