A338708 - OEIS

%I #18 Jan 26 2025 21:47:37

%S 0,0,0,0,0,0,0,0,0,1,6,37,158,591,1896,5537,14812,37133,87841,198267,

%T 429199,896731,1814978,3572810,6858774,12874977,23679669,42752787,

%U 75887244,132618635,228443753,388297169,651868064,1081771385,1775876764,2885944062,4645393253,7410678577,11722238660,18394159344

%N Number of 4-linear trees on n nodes.

%H Andrew Howroyd, <a href="/A338708/b338708.txt">Table of n, a(n) for n = 1..1000</a>

%H Tanay Wakhare, Eric Wityk, and Charles R. Johnson, <a href="https://doi.org/10.1016/j.disc.2020.112008">The proportion of trees that are linear</a>, Discrete Mathematics 343.10 (2020): 112008. Also <a href="https://doi.org/10.1016/j.disc.2020.112284">Corrigendum</a> and preprint <a href="https://arxiv.org/abs/1901.08502">arXiv:1901.08502 [math.CO]</a>, 2019-2020. See Tables 1 and 2 (but beware errors).

%F G.f.: x^4*((P(x) - 1/(1 - x))^2*(P(x) - 1)^2/(1 - x)^3 + (P(x^2) - 1/(1 - x^2))*(P(x^2) - 1)/((1 - x^2)*(1 - x)))/2 where P(x) is the g.f. of A000041. - _Andrew Howroyd_, Jan 26 2025

%o (PARI) seq(n)={my(A=O(x^(n-5)), p=1/eta(x + A), p2=1/eta(x^2 + A)); Vec(((p - 1/(1-x))^2*(p - 1)^2/(1 - x)^3 + (p2 - 1/(1 - x^2))*(p2 - 1)/((1 - x^2)*(1 - x)))/2, -n)} \\ _Andrew Howroyd_, Jan 26 2025

%Y Column k=4 of A380363.

%Y Cf. A000041, A130131, A338706, A338707.

%K nonn

%O 1,11

%A _N. J. A. Sloane_, Nov 05 2020

%E a(26) onwards from _Andrew Howroyd_, Jan 26 2025