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A254314 Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).

%I A254314 #32 Mar 14 2024 05:59:42
%S A254314 1,1,2,5,14,43,143,507,1887,7279,28828,116455,477709,1983779,8321474,
%T A254314 35203777,150014157,643302743,2773997104,12020733635,52319374842,
%U A254314 228616865437,1002544803949,4410700121313,19462407890220,86111960348939,381956399941011
%N A254314 Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).
%C A254314 a(n+1) is the number of rooted plane trees with integer compositions labeling the leaves (empty labels are allowed), with total size n. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling the leaves.
%C A254314 Example: a(3)=5 because there are 5 elements of size 2: two trees that consist of the root and no descendants, hence the root is itself a leaf and it can be labeled by either 2=2 or by 1=1+1, then a tree with the root and one descendant that is a leaf labeled with 1=1, then a tree with the root and two descendants with no labels on the leaves, and finally a tree with the root with one descendant with a descendant that is a leaf with no label. - _Ricardo Gómez Aíza_, Feb 29 2024
%H A254314 G. C. Greubel, <a href="/A254314/b254314.txt">Table of n, a(n) for n = 0..1000</a>
%H A254314 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. 17-18.
%F A254314 G.f. A(x) satisfies 0 = (2*x-1)*A(x)^2 + (x^2-6*x+3)*A(x) + (3*x-2).
%F A254314 G.f.: (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*(1 - 2*x)).
%F A254314 Conjecture: n*a(n) +2*(-5*n+6)*a(n-1) +2*(17*n-39)*a(n-2) +6*(-8*n+27)*a(n-3) +(25*n-114)*a(n-4) +2*(-n+6)*a(n-5)=0. - _R. J. Mathar_, Jun 07 2016
%F A254314 a(n) ~ sqrt(b*(5-32*b+46*b^2))/(2*sqrt((1-2*b)^3*Pi*n^3))*(1/b)^n where b = (11-c-100/c)/3 and c = (-998+6*sqrt(111)*i)^(1/3). - _Ricardo Gómez Aíza_, Feb 29 2024
%e A254314 G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 143*x^6 + 507*x^7 + ...
%t A254314 CoefficientList[Series[(3-6*x+x^2 - Sqrt[(1-4*x+x^2)^2 -4*x^3])/(2*(1 - 2*x)), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 10 2018 *)
%o A254314 (PARI) {a(n) = if( n<0, 0, polcoeff( (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 + x^2 * O(x^n))) / (2*(1 - 2*x)), n))};
%o A254314 (Magma) m:=60; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-6*x+x^2 - Sqrt((1-4*x+x^2)^2 -4*x^3))/(2*(1 - 2*x)))); // _G. C. Greubel_, Aug 10 2018
%Y A254314 Cf. A006720.
%K A254314 nonn
%O A254314 0,3
%A A254314 _Michael Somos_, Jan 28 2015