A338706 Number of 2-linear trees on n nodes.

0, 0, 0, 0, 0, 1, 3, 10, 24, 56, 114, 224, 411, 733, 1252, 2091, 3393, 5408, 8440, 12982, 19650, 29388, 43394, 63430, 91754, 131584, 187057, 263932, 369624, 514253, 710838, 976876, 1334828, 1814492, 2454011, 3303436, 4426627, 5906599, 7848883, 10389557

Offset

1,7

Comments

A k-linear tree is a tree with exactly k vertices of degree 3 or higher all of which lie on a path. - Andrew Howroyd, Dec 17 2020

Empirically the partial sums of A000147. - Sean A. Irvine, Jul 11 2022

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Tanay Wakhare, Eric Wityk, and Charles R. Johnson, The proportion of trees that are linear, Discrete Mathematics 343.10 (2020): 112008. Also Corrigendum and preprint arXiv:1901.08502 [math.CO], 2019-2020. See Tables 1 and 2 (but beware errors).

Formula

G.f.: ((x*(P(x) - 1/(1-x)))^2 + x^2*(P(x^2) - 1/(1-x^2)))/(2*(1-x)) where P(x) is the g.f. of A000041. - Andrew Howroyd, Dec 17 2020

Example

The a(6) = 1 tree is:
         o   o
         |   |
     o---o---o---o

Prog

(PARI) seq(n)=my(p=1/(eta(x + O(x^(n-3))))); Vec(((x*(p - 1/(1-x)))^2 + x^2*(subst(p, x, x^2) - 1/(1-x^2)))/(2*(1-x)), -n) \\ Andrew Howroyd, Dec 17 2020

Crossrefs

Column k=2 of A380363 and A238415.

Cf. A000041, A000147, A130131, A338707, A338708.

Sequence in context: A062446 A053208 A355024 * A338710 A336516 A286209

Adjacent sequences: A338703 A338704 A338705 * A338707 A338708 A338709

Keyword

nonn

Author

N. J. A. Sloane, Nov 05 2020, using data supplied by Eric Wityk

Extensions

Terms a(31) and beyond from Andrew Howroyd, Dec 17 2020

Status

approved