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A026571 a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.

%I A026571 #37 Mar 14 2024 06:00:35
%S A026571 1,2,7,16,44,106,273,672,1696,4214,10573,26392,66151,165578,415277,
%T A026571 1041480,2615004,6568450,16512355,41531360,104526093,263206638,
%U A026571 663143211,1671581968,4215574482,10635988422,26846320149,67790042264
%N A026571 a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.
%C A026571 From _Ricardo Gómez Aíza_, Feb 26 2024: (Start)
%C A026571 The sequence corresponds to the cumulative distribution function of the number of petals in a rooted plane tree with nonempty flowers everywhere but on the root, with flowers made out of petals of size one.
%C A026571 Examples:
%C A026571 a(2)=1 because there is only one element of size 2, and it consists of the root with one descendant with a flower with a single petal attached to it;
%C A026571 a(3)=2 because again there is only one element of size 3 that consists of the root with one descendant with a flower with two petals attached to it;
%C A026571 a(4)=7 because there is one tree with the root and two descendants, each with a flower with one petal only (two petals in total), then there is one tree with the root and one descendant that also has a descendant, and both descendants with a flower with one petal only (two petals in total), and finally there is the tree with the root and one descendant with a flower with three petals. (End)
%H A026571 Michael De Vlieger, <a href="/A026571/b026571.txt">Table of n, a(n) for n = 2..2450</a>
%H A026571 Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 18-19.
%F A026571 Conjecture: (n+2)*a(n) + 3*(-n-1)*a(n-1) - 3*n*a(n-2) + 11*(n-1)*a(n-3) + 2*(n-6)*a(n-4) + 4*(-2*n+7)*a(n-5) = 0. - _R. J. Mathar_, Jun 23 2013
%F A026571 From _Ricardo Gómez Aíza_, Feb 26 2024: (Start)
%F A026571 G.f.: (2*x^2+x-1+(1-x)*p(x))/(2*(x^3-x^2)*p(x)) with p(x) = sqrt((1-x-4*x^2)/(1-x)).
%F A026571 a(n) ~ 16*sqrt((9-s)/(s*(s-1)^5*Pi*n))*(8/(s-1))^n where s=sqrt(17). (End)
%t A026571 CoefficientList[Series[(2*x^2 + x - 1 + (1 - x)*#)/(2*(x^3 - x^2)*#) &[Sqrt[(1 - x - 4*x^2)/(1 - x)]], {x, 0, 29}], x] (* _Michael De Vlieger_, Mar 03 2024 *)
%K A026571 nonn
%O A026571 2,2
%A A026571 _Clark Kimberling_