A338707 Number of 3-linear trees on n nodes.

0, 0, 0, 0, 0, 0, 0, 1, 5, 22, 74, 219, 576, 1394, 3150, 6733, 13744, 26969, 51185, 94323, 169453, 297533, 512006, 865050, 1437739, 2353756, 3801041, 6060918, 9552826, 14894428, 22991659, 35159606, 53299703, 80137271, 119563216, 177091225, 260504790, 380720841

Offset

1,9

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

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 on arXiv, arXiv:1901.08502 [math.CO], 2019-2020. See Tables 1 and 2 (but beware errors).

Formula

G.f.: x^3*(P(x)-1)*((P(x) - 1/(1-x))^2/(1-x)^2 + (P(x^2) - 1/(1-x^2))/(1-x^2))/2 where P(x) is the g.f. of A000041. - Andrew Howroyd, Dec 17 2020

Prog

(PARI) seq(n)={my(p=1/(eta(x + O(x^(n-5))))); Vec(x^3*(p-1)*((p - 1/(1-x))^2/(1-x)^2 + (subst(p, x, x^2) - 1/(1-x^2))/(1-x^2))/2, -n)} \\ Andrew Howroyd, Dec 17 2020

Crossrefs

Cf. A000041, A130131, A338706.

Sequence in context: A011846 A241694 A220733 * A058750 A058752 A234351

Adjacent sequences: A338704 A338705 A338706 * A338708 A338709 A338710

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