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Number of exceptional sets of roots of type D_n. Also the number of unordered factorizations of the Coxeter element.
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%I #16 Feb 29 2024 17:46:42

%S 12,87,584,3835,25008,162792,1060048,6910695,45119100,295038315,

%T 1932260256,12673336052,83236707232,547388545740,3604063891104,

%U 23755630474079,156740823815940,1035157282013085,6842413166034600,45265133475699795,299671339559444160,1985322768625822080

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

%H F. Chapoton <a href="https://irma.math.unistra.fr/~chapoton/clusters.html">Cluster algebras</a>

%F 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).

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

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

%o (MuPAD)

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

%o (Sage)

%o def A137207(n):

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

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

%K nonn

%O 3,1

%A _F. Chapoton_, Mar 05 2008

%E a(22)-a(24) from _Stefano Spezia_, Feb 29 2024