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 arXiv:1901.08502v2. 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
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