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A338706 Number of 2-linear trees on n nodes.

%I A338706 #23 Jan 26 2025 21:47:48
%S A338706 0,0,0,0,0,1,3,10,24,56,114,224,411,733,1252,2091,3393,5408,8440,
%T A338706 12982,19650,29388,43394,63430,91754,131584,187057,263932,369624,
%U A338706 514253,710838,976876,1334828,1814492,2454011,3303436,4426627,5906599,7848883,10389557
%N A338706 Number of 2-linear trees on n nodes.
%C A338706 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
%C A338706 Empirically the partial sums of A000147. - _Sean A. Irvine_, Jul 11 2022
%H A338706 Andrew Howroyd, <a href="/A338706/b338706.txt">Table of n, a(n) for n = 1..1000</a>
%H A338706 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 A338706 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
%e A338706 The a(6) = 1 tree is:
%e A338706          o   o
%e A338706          |   |
%e A338706      o---o---o---o
%o A338706 (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
%Y A338706 Column k=2 of A380363 and A238415.
%Y A338706 Cf. A000041, A000147, A130131, A338707, A338708.
%K A338706 nonn
%O A338706 1,7
%A A338706 _N. J. A. Sloane_, Nov 05 2020, using data supplied by Eric Wityk
%E A338706 Terms a(31) and beyond from _Andrew Howroyd_, Dec 17 2020