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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.
1
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
0,5
COMMENTS
First differs from A000055 at a(10).
First differs from A130131 at a(10), n >= 1.
LINKS
FORMULA
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
EXAMPLE
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)
PROG
(PARI) 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
CROSSREFS
Sequence in context: A359392 A199142 A090344 * A389602 A198662 A198620
KEYWORD
nonn
AUTHOR
Gabriel Burns, Oct 31 2016
EXTENSIONS
Corrections and more terms from Andrey Zabolotskiy, Nov 21 2016
a(24) onwards from Andrew Howroyd, Feb 06 2025
STATUS
approved