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.

1, 2, 7, 16, 44, 106, 273, 672, 1696, 4214, 10573, 26392, 66151, 165578, 415277, 1041480, 2615004, 6568450, 16512355, 41531360, 104526093, 263206638, 663143211, 1671581968, 4215574482, 10635988422, 26846320149, 67790042264

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

From Ricardo Gómez Aíza, Feb 26 2024: (Start)

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.

Examples:

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;

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;

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)

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..2450
• 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. 18-19.

Programs

Mathematica

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 *)

Formula

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

From Ricardo Gómez Aíza, Feb 26 2024: (Start)

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)).

a(n) ~ 16*sqrt((9-s)/(s*(s-1)^5*Pi*n))*(8/(s-1))^n where s=sqrt(17). (End)