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%I M1448 #56 Jul 01 2021 03:30:32
%S 1,1,2,5,13,37,111,345,1105,3624,12099,41000,140647,487440,1704115,
%T 6002600,21282235,75890812,272000538,979310627,3540297130,12845634348,
%U 46764904745,170767429511,625314778963,2295635155206,8447553316546,31153444946778,115122389065883
%N Number of rooted triangular cacti with 2n+1 nodes (n triangles).
%C a(n) is also the number of isomorphism classes of Fano Bott manifolds of complex dimension n (see [Cho-Lee-Masuda-Park]). - _Eunjeong Lee_, Jun 29 2021
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 305, (4.2.34).
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.20).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003080/b003080.txt">Table of n, a(n) for n = 0..1000</a>
%H Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.
%H Yunhyung Cho, Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, <a href="https://arxiv.org/abs/2106.12788">On the enumeration of Fano Bott manifolds</a>, arXiv:2106.12788 [math.AG], 2021. See Table 1 p. 8.
%H P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/volumes/16-1/PDF/053-080.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
%H P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F a(n)=b(2n+1). b shifts left under transform T where Tb = EULER(E_2(b)). E_2(b) has g.f. (B(x^2)+B(x)^2)/2.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.90053254788870206167147120260433375638561926371844809... and c = 0.4861961460367182791173441493565088408563977498871021... - _Vaclav Kotesovec_, Jul 01 2021
%t terms = 30;
%t nmax = 2 terms;
%t A[_] = 0; Do[A[x_] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
%t DeleteCases[CoefficientList[A[x], x], 0] (* _Jean-François Alcover_, Sep 02 2018 *)
%Y Column k=3 of A332648.
%Y Cf. A003081, A034940, A034941.
%K nonn,eigen,nice
%O 0,3
%A _N. J. A. Sloane_
%E Sequence extended by _Paul Zimmermann_, Mar 15 1996
%E Additional comments from _Christian G. Bower_
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