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A130131
Number of n-lobsters.
13
1, 1, 1, 2, 3, 6, 11, 23, 47, 105, 231, 532, 1224, 2872, 6739, 15955, 37776, 89779, 213381, 507949, 1209184, 2880382, 6861351, 16348887, 38955354, 92831577, 221219963, 527197861, 1256385522, 2994200524, 7135736613, 17005929485, 40528629737, 96588403995, 230190847410
OFFSET
1,4
COMMENTS
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
LINKS
G. Li and F. Ruskey, C program [dead link]
Tanay Wakhare, Eric Wityk, and Charles R. Johnson, The proportion of trees that are linear, Discrete Mathematics 343.10 (2020): 112008. Also Corrigendum and preprint arXiv:1901.08502. See Tables 1 and 2 (but beware errors).
Eric Weisstein's World of Mathematics, Lobster Graph
EXAMPLE
a(10) = 105 = A000055(10) - 1 because all trees with 10 vertices are lobsters except this one:
o-o-o
/
o-o-o-o
\
o-o-o
Also, all trees with 10 vertices are linear (all vertices of degree >2 belong to a single path) except this one:
o o
\ /
o
|
o
/ \
o o
/ \ / \
o o o o
MATHEMATICA
eta = QPochhammer;
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))];
CoefficientList[s[32], x] // Rest (* Jean-François Alcover, Nov 17 2020, after Andrew Howroyd *)
PROG
(PARI)
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))}
Vec(s(30)) \\ Andrew Howroyd, Nov 02 2017
CROSSREFS
Row sums of A380363.
Cf. k-linear trees for k = 1..4: A004250, A338706, A338707, A338708.
Sequence in context: A198655 A211694 A303586 * A338355 A277796 A123465
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, May 11 2007
EXTENSIONS
a(15)-a(32) from Washington Bomfim, Feb 23 2011
STATUS
approved