A130131 -id:A130131 - cp's OEIS frontend

A186308 Number of lobsters with n nodes that are not caterpillars.

0, 0, 0, 0, 1, 3, 11, 33, 95, 260, 696, 1816, 4659, 11795, 29520, 73267, 180485, 442157, 1077856, 2617726, 6336551, 15299287, 36857178, 88635225, 212829307, 510416549, 1222826994, 2927083468, 7001510693, 16737477645

Offset: 3

Washington Bomfim, Feb 23 2011

nonn

			All the trees of order less than 7 are caterpillars. Only one tree with 7 nodes is a lobster and is not a caterpillar, so a(3)=a(4)=a(5)=a(6)=0, and a(7)=1.

Wikipedia, Caterpillar tree

Cf. A000055, A130131.

a(n) = A130131(n) - (2^(n-4) + 2^[(n-4)/2]).

A277795 Number of trees with n unlabeled nodes such that all nodes with degree >2 lie on a single path with length equal to the tree's diameter.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 103, 223, 503, 1132, 2602, 5986, 13922, 32433, 75994, 178354, 419945, 990134, 2339033, 5531459, 13097217, 31036235, 73607165, 174677138, 414768535, 985315906, 2341687487, 5567158277, 13239573207, 31494089609, 74935197166, 178332248260, 424473745066

Offset: 0

Gabriel Burns, Oct 31 2016

nonn

First differs from A000055 at a(10).

First differs from A130131 at a(10), n >= 1.

			From _Andrey Zabolotskiy_, Nov 21 2016: (Start)
Three trees that are counted in A000055(10) but not in a(10):
(1)
  o  o-o-o
  |    |
  o----o
  |    |
  o  o-o-o
(2)
  o-o-o
    |
    o-o-o-o
    |
  o-o-o
(3)
  o-o-o-o-o-o-o
        |
      o-o-o
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Andrey Zabolotskiy, Python script for the sequence

Cf. A000055, A130131.

seq(n)={my(s=1+x, p=1+O(x^n), p2=p, q=p, q2=p); for(k=1, n\2, q*=p^2; q2*=p2; p /= 1-x^k; p2 /= 1-x^(2*k); s+=x^(2*k)*(q+q2)*(1+x*p)/2); Vec(s+O(x*x^n))} \\ Andrew Howroyd, Feb 06 2025

G.f.: Sum_{k>=0} x^(2*k)*(Q(k,x)^2 + Q(k,x^2))*(1 + x*P(k,x))/2, where P(x,k) = 1/Product_{i=1..k} (1-x^i) and Q(x,k) = 1/Product_{i=1..k-1} (1-x^i)^(k-i). - Andrew Howroyd, Feb 06 2025

Corrections and more terms from Andrey Zabolotskiy, Nov 21 2016

a(24) onwards from Andrew Howroyd, Feb 06 2025

cp's OEIS Frontend

A186308 Number of lobsters with n nodes that are not caterpillars.

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A277795 Number of trees with n unlabeled nodes such that all nodes with degree >2 lie on a single path with length equal to the tree's diameter.

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