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

1, 1, 2, 5, 14, 43, 143, 507, 1887, 7279, 28828, 116455, 477709, 1983779, 8321474, 35203777, 150014157, 643302743, 2773997104, 12020733635, 52319374842, 228616865437, 1002544803949, 4410700121313, 19462407890220, 86111960348939, 381956399941011

Offset: 0

Author

Michael Somos, Jan 28 2015

Keywords

• nonn

Comments

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.

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

Examples

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 143*x^6 + 507*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 17-18.

Crossrefs

Cf. A006720.

Programs

Magma

m:=60; R:=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

Mathematica

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 *)

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))};

Formula

G.f. A(x) satisfies 0 = (2*x-1)*A(x)^2 + (x^2-6*x+3)*A(x) + (3*x-2).

G.f.: (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*(1 - 2*x)).

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

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