Number of Lobsters
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Let P(x) be the generating function for the number of integer partitions.

Case diameter>4
A lobster of diameter greater than 4 consists of an end cell followed by zero or more middle cells followed by an end cell.
Reversals of the sequence are not considered distinct.
An end cell consists of a rooted tree with height exactly 2 whilst a middle cell consists of a rooted tree with height at most 2.

Let M(x) be the generating function for a middle cell.
A rooted tree of height at most 2 can be split into zero or more child trees of height at most 1.
The number of rooted trees on n nodes of height at most 1 is just 1.
Therefore the number of rooted trees on n nodes of height at most 2 is just the number of integer partitions of n-1.

M(x) = x*P(x).

Let E(x) be the generating function for an end cell:
The case of a rooted tree of height at most 1 needs to be excluded.

E(x) = x*P(x) - x/(1-x) = x*(P(x) - 1/(1-x)).

Let S(x) be the generating function for a sequence of middle cells:

S(x) = 1/(1-M(x)).

Let B(x) be the generating function for the number of lobsters of diameter > 4:

B(x) = (E(x)^2 * S(x) + E(x^2) * S(x^2) * (1 + M(x))) / 2.

Case diameter<=4
All trees with diameter at most 4 are lobsters.
There is already a sequence for this: A058984.

Let C(x) be the generating function for the number of lobsters of diameter <= 4:

C(x) = x*P(x) - x^3/((1-x)^2*(1+x)). 

Overall formula
Let A(x) be the generating function for the number of lobsters

A(x) = B(x) + C(x)
     = x^2*((P(x) - 1/(1-x))^2/(1-x*P(x)) + (P(x^2) - 1/(1-x^2))*(1 + x*P(x))/(1-x^2*P(x^2)))/2 + x*P(x) - x^3/((1-x)^2*(1+x)).