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

Offset: 3

Author

F. Chapoton, Mar 05 2008

Keywords

• nonn

Examples

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

Links

• F. Chapoton Cluster algebras

Crossrefs

Cf. A001764 for type A, A045721 for type B.

Programs

MuPAD

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

Sage

def A137207(n):
    return (16*n*n-41*n+24)*binomial(3*n-5,n-2)/n/(2*n-1)

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).

Extensions

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