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A376161 Number of support Tau-tilting modules for some algebras.

%I A376161 #45 Apr 05 2025 10:12:09
%S A376161 3,5,12,33,98,306,990,3289,11154,38454,134368,474810,1693812,6091780,
%T A376161 22064130,80410185,294647250,1084922190,4012165080,14895504030,
%U A376161 55496654460,207431394300,777601790940,2922867908298,11013796950228,41596652545756,157434454904160,597029454416724,2268232385053096
%N A376161 Number of support Tau-tilting modules for some algebras.
%C A376161 See Prop. A.6 in Wang's reference for the table counting Tau-tilting modules for the linear quiver modulo the relation alpha*beta = 0.
%H A376161 Paolo Xausa, <a href="/A376161/b376161.txt">Table of n, a(n) for n = 0..1000</a>
%H A376161 Anna Rodriguez Rasmussen, <a href="https://arxiv.org/abs/2504.01706">Exact Borel subalgebras of quasi-hereditary monomial algebras</a>, arXiv:2504.01706 [math.RT], 2025. See p. 38.
%H A376161 Qi Wang, <a href="https://arxiv.org/abs/1910.01937">Tau-tilting finite simply connected algebras</a>, arXiv:1910.01937 [math.RT], 2019-2022.
%F A376161 a(n) = 3*(3*n+2)*binomial(2*n+4,n+2)/(4*(2*n+1)*(2*n+3)).
%F A376161 a(n) = A329533(n)/(n + 1).
%F A376161 From _Peter Luschny_, Sep 13 2024: (Start)
%F A376161 a(n) = (3*n + 2) * [x^n] ((1 - 4*x)^(3/2) + 12*x - 2)/(4*x^2).
%F A376161 a(n) = A016789(n)*(3/2)*(2*n)! * [x^(2*n)] hypergeom([], [3], x^2).
%F A376161 a(n) = CatalanNumber(n)*(9*n + 6)/(n + 2).
%F A376161 a(n) = -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2).
%F A376161 a(n) = 2^n*(9*n + 6)*(2*n - 1)!! / (n + 2)!.
%F A376161 a(n) = A007054(n) * (3*n + 2) / 2.
%F A376161 a(n) = 6*A023999(n + 1)/(n + 2)!. (End)
%p A376161 a := n -> -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2):
%p A376161 seq(a(n), n = 0..28)  # _Peter Luschny_, Sep 13 2024
%t A376161 A376161[n_] := CatalanNumber[n]*(9*n + 6)/(n + 2);
%t A376161 Array[A376161, 30, 0] (* _Paolo Xausa_, Sep 14 2024 *)
%o A376161 (Sage)
%o A376161 def a(n):
%o A376161     return 3*(3*n+2)*binomial(2*n+4,n+2)/4/(2*n+1)/(2*n+3)
%Y A376161 Cf. A005807, A007054, A000108, A016789, A023999, A329533.
%K A376161 nonn
%O A376161 0,1
%A A376161 _F. Chapoton_, Sep 13 2024