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A380256 Number of rooted binary normal unlabeled galled trees with n leaves and exactly 1 gall.

%I A380256 #16 Jan 19 2025 12:58:14
%S A380256 0,0,0,1,4,15,48,148,435,1250,3512,9726,26587,71975,193200,515051,
%T A380256 1364896,3598794,9447028,24704031,64382465,167288460,433512724,
%U A380256 1120719444,2891035926,7443225226,19129208972,49082742607,125752279124,321744111359,822165920924,2098475215237
%N A380256 Number of rooted binary normal unlabeled galled trees with n leaves and exactly 1 gall.
%C A380256 The asymptotic growth of a(n) follows (0.3910...)(2.4833...^n)n^(1/2), where 2.4833... is the constant represented by A086317.
%H A380256 Lily Agranat-Tamir, Shaili Mathur, and Noah A. Rosenberg, <a href="https://doi.org/10.1007/s11538-024-01270-8">Enumeration of rooted binary unlabeled galled trees</a>, Bull. Math. Biol. 86 (2024), 45. (see Table 3)
%H A380256 Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg, <a href="https://doi.org/10.4230/LIPIcs.AofA.2024.27">Asymptotic enumeration of rooted binary unlabeled galled trees with a fixed number of galls</a>. In C. Mailler, S. Wild, eds. Proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs) 302: 27. Schloss Dagstuhl — Leibniz-Zentrum für Informatik.
%F A380256 G.f.: 1/(1-U(x)) - 1/(1-U(x))^2 + U(x)/(2*(1-U(x))^3) + U(x)/(2*(1-U(x))*(1-U(x^2))), where U(x) is the g.f. of A001190 (eq. 48 of Agranat-Tamir et al., Bull. Math. Biol. 86 (2024), 45).
%e A380256 For n=3 leaves, there is a unique rooted binary unlabeled tree with a root gall from which 3 leaves are descended; hence a(3)=1. This galled tree has the shape:
%e A380256      .
%e A380256     / \
%e A380256    ._._.
%e A380256   /  |  \
%Y A380256 Cf. A001190 (rooted binary unlabeled galled trees with n leaves and 0 galls), A380211 (rooted binary unlabeled galled trees with n leaves and any number of galls).
%Y A380256 Radius of convergence of the generating function follows the contstant A240943 (exponential growth according to A086317).
%K A380256 nonn
%O A380256 0,5
%A A380256 _Noah A Rosenberg_, Jan 17 2025