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A189942 Table, read by rows, of the number of quivers of type Ã_(n-1) according to the parameter k (n >= 2, 1 <= k <= [n/2]).

%I A189942 #29 Jan 19 2022 14:30:39
%S A189942 1,2,5,4,14,12,42,36,22,132,108,100,429,349,315,172,1430,1144,1028,
%T A189942 980,4862,3868,3432,3240,1651,16796,13260,11700,10920,10584,58786,
%U A189942 46210,40520,37556,36036,18028,208012,162792,142120,130900,124740,121968
%N A189942 Table, read by rows, of the number of quivers of type Ã_(n-1) according to the parameter k (n >= 2, 1 <= k <= [n/2]).
%C A189942 Table 1, p. 15 of Bastian.
%C A189942 There is a bijection with dissections of an annulus [Hermund André Torkildsen]. - _N. J. A. Sloane_, Jan 31 2013
%D A189942 Francois Bergeron, Gilbert Labelle and Pierre Leroux, Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, Cambridge, 1998, Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota.
%H A189942 Ibrahim Assem, Thomas Brustle, Gabrielle Charbonneau-Jodoin and Pierre-Guy Plamondon, <a href="http://arxiv.org/abs/0903.3347">Gentle algebras arising from surface triangulations</a>, Algebra & Number Theory 4 (2010), no. 2, 201-229; arXiv:0903.3347 [math.RT], 2009.
%H A189942 Janine Bastian, Thomas Prellberg, Martin Rubey and Christian Stump, <a href="http://arxiv.org/abs/0906.0487">Counting the number of elements in the mutation classes of Ã_n-quivers</a>, arXiv:0906.0487 [math.CO], 2009-2011.
%H A189942 Janine Bastian, <a href="http://arxiv.org/abs/0901.1515">Mutation classes of Ã_n-quivers and derived equivalence classification of cluster tilted algebras of type Ã_n</a>, Algebra Number Theory 5 (2011), no. 5, 567-594; arXiv:0901.1515 [math.RT], 2009-2012.
%H A189942 Hermund André Torkildsen, <a href="http://arxiv.org/abs/1208.2138">A geometric realization of the m-cluster category of type Ã</a>, arXiv 1208.2138 [math.RT], 2012. - From _N. J. A. Sloane_, Jan 31 2013
%e A189942 The table begins
%e A189942 ===================================
%e A189942 n  |  r=1 | r=2 | r=3 | r=4 | r=5 |
%e A189942 ===================================
%e A189942 n=2     1
%e A189942 n=3     2
%e A189942 n=4     5      4
%e A189942 n=5    14     12
%e A189942 n=6    42     36    22
%e A189942 n=7   132    108   100
%e A189942 n=8   429    349   315   172
%e A189942 n=9  1430   1144  1028   980
%e A189942 n=10 4862   3868  3432  3240  1651
%e A189942 ===================================
%t A189942 a[r_, r_] := 1/2 (Binomial[2 r, r]/2 + Sum[EulerPhi[k]/(4 r) Binomial[2 r/k, r/k]^2, {k, Divisors@r}]);
%t A189942 a[r_, s_] := 1/2 Sum[EulerPhi[k]/(r + s) Binomial[2 r/k, r/k] Binomial[2 s/k, s/k], {k, Intersection[Divisors@r, Divisors@s]}];
%t A189942 Table[a[r, n - r], {n, 2, 10}, {r, n/2}] // TableForm
%t A189942 (* _Andrey Zabolotskiy_, Jan 19 2022 *)
%Y A189942 Cf. A000108 (r=1), A000888 (n=2r+1).
%K A189942 nonn,tabf
%O A189942 2,2
%A A189942 _Jonathan Vos Post_, May 01 2011
%E A189942 Rows 11-13 added by _Andrey Zabolotskiy_, Jan 19 2022