|
%I #32 Mar 14 2024 05:59:42
%S 1,1,2,5,14,43,143,507,1887,7279,28828,116455,477709,1983779,8321474,
%T 35203777,150014157,643302743,2773997104,12020733635,52319374842,
%U 228616865437,1002544803949,4410700121313,19462407890220,86111960348939,381956399941011
%N Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).
%C 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 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 G. C. Greubel, <a href="/A254314/b254314.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 G.f. A(x) satisfies 0 = (2*x-1)*A(x)^2 + (x^2-6*x+3)*A(x) + (3*x-2).
%F G.f.: (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*(1 - 2*x)).
%F 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 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 G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 143*x^6 + 507*x^7 + ...
%t 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 (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 (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 Cf. A006720.
%K nonn
%O 0,3
%A _Michael Somos_, Jan 28 2015
| 