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A380306 Irregular triangle read by rows: T(n,k) is the number of rooted binary normal unlabeled galled trees with n leaves and exactly k galls, 0 <= k <= floor((n-1)/2).

%I A380306 #4 Feb 01 2025 23:12:16
%S A380306 1,1,1,1,2,4,3,15,2,6,48,18,11,148,107,6,23,435,528,78,46,1250,2295,
%T A380306 661,19,98,3512,9185,4356,346,207,9726,34503,24564,3776,67,451,26587,
%U A380306 123612,123825,31289,1543,983,71975,426218,574149,216501,20720,246
%N A380306 Irregular triangle read by rows: T(n,k) is the number of rooted binary normal unlabeled galled trees with n leaves and exactly k galls, 0 <= k <= floor((n-1)/2).
%C A380306 For fixed k, the asymptotic growth of T(n,k) with n follows (2^(2*k-1) / ((2*k)! * g^(4*k-1) * sqrt(Pi))) * n^(2*k-3/2) * r^n, where r is the constant 2.4833... represented by A086317 and g is a constant 1.1300... (Theorem 10 of Agranat-Tamir et al., Leibniz International Proceedings in Informatics (LIPIcs) 302 (2024), 27).
%H A380306 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 A380306 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 A380306 G.f. satisfies A(x,y) = x + y + (1/2)*A(x,y)^2 + (1/2)*A(x^2,y^2) - y/(1-A(x,y)) + y*A(x,y)/(2*(1-A(x,y))^2) + y*A(x,y)/(2*(1-A(x^2,y^2))) (eq. 56 of Agranat-Tamir et al., Bull. Math. Biol. 86 (2024), 45).
%e A380306 Triangle begins:
%e A380306      1;
%e A380306      1;
%e A380306      1,      1;
%e A380306      2,      4;
%e A380306      3,     15,       2;
%e A380306      6,     48,      18;
%e A380306     11,    148,     107,       6;
%e A380306     23,    435,     528,      78;
%e A380306     46,   1250,    2295,     661,      19;
%e A380306     98,   3512,    9185,    4356,     346;
%e A380306    207,   9726,   34503,   24564,    3776,     67;
%e A380306    451,  26587,  123612,  123825,   31289,   1543;
%e A380306    983,  71975,  426218,  574149,  216501,  20720,  246;
%e A380306   2179, 193200, 1425011, 2493129. 1316450, 206644, 6942;
%Y A380306 First column (k=0) is A001190.
%Y A380306 Second column (k=1) is A380256.
%Y A380306 Row sums give A380211.
%K A380306 nonn,tabf
%O A380306 1,5
%A A380306 _Noah A Rosenberg_, Jan 19 2025