%I #41 Aug 05 2024 06:02:13

%S 1,1,1,2,3,6,11,23,47,105,231,532,1224,2872,6739,15955,37776,89779,

%T 213381,507949,1209184,2880382,6861351,16348887,38955354,92831577,

%U 221219963,527197861,1256385522,2994200524,7135736613,17005929485

%N Number of n-lobsters.

%C A lobster graph is a tree having the property that the removal of all leaf nodes leaves a caterpillar graph (see A005418). - _N. J. A. Sloane_, Nov 05 2020

%H Andrew Howroyd, <a href="/A130131/b130131.txt">Table of n, a(n) for n = 1..200</a>

%H Andrew Howroyd, <a href="/A130131/a130131.txt">Formula for number of lobsters</a>

%H G. Li and F. Ruskey, <a href="http://theory.cs.uvic.ca/dis/distribute.pl.cgi?package=FreeTree.c">C program</a>

%H Wakhare, Tanay, Eric Wityk, and Charles R. Johnson. "The proportion of trees that are linear." Discrete Mathematics 343.10 (2020): 112008. Also <a href="https://arxiv.org/abs/1901.08502">arXiv:1901.08502v2</a>. See Tables 1 and 2 (but beware errors).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LobsterGraph.html">Lobster Graph</a>

%t eta = QPochhammer;

%t s[n_] := With[{ox = O[x]^n}, x^2 ((1/eta[x + ox] - 1/(1 - x))^2/(1 - x/eta[x + ox]) + (1/eta[x^2 + ox] - 1/(1 - x^2))(1 + x/eta[x + ox])/(1 - x^2/eta[x^2 + ox]))/2 + x/eta[x + ox] - x^3/((1 - x)^2*(1 + x))];

%t CoefficientList[s[32], x] // Rest (* _Jean-François Alcover_, Nov 17 2020, after _Andrew Howroyd_ *)

%o (PARI)

%o s(n)={my(ox=O(x^n)); x^2*((1/eta(x+ox)-1/(1-x))^2/(1-x/eta(x+ox)) + (1/eta(x^2+ox)-1/(1-x^2))*(1+x/eta(x+ox))/(1-x^2/eta(x^2+ox)))/2 + x/eta(x+ox) - x^3/((1-x)^2*(1+x))}

%o Vec(s(30)) \\ _Andrew Howroyd_, Nov 02 2017

%Y Cf. A000055, A005418, A058984, A130132, A331693.

%K nonn

%O 1,4

%A _Eric W. Weisstein_, May 11 2007

%E a(15)-a(32) from _Washington Bomfim_, Feb 23 2011