cp's OEIS Frontend

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.

%I A277795 #24 Feb 07 2025 19:37:42
%S A277795 1,1,1,1,2,3,6,11,23,47,103,223,503,1132,2602,5986,13922,32433,75994,
%T A277795 178354,419945,990134,2339033,5531459,13097217,31036235,73607165,
%U A277795 174677138,414768535,985315906,2341687487,5567158277,13239573207,31494089609,74935197166,178332248260,424473745066
%N 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.
%C A277795 First differs from A000055 at a(10).
%C A277795 First differs from A130131 at a(10), n >= 1.
%H A277795 Andrew Howroyd, <a href="/A277795/b277795.txt">Table of n, a(n) for n = 0..1000</a>
%H A277795 Andrey Zabolotskiy, <a href="/A277795/a277795.py.txt">Python script for the sequence</a>
%F A277795 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
%e A277795 From _Andrey Zabolotskiy_, Nov 21 2016: (Start)
%e A277795 Three trees that are counted in A000055(10) but not in a(10):
%e A277795 (1)
%e A277795   o  o-o-o
%e A277795   |    |
%e A277795   o----o
%e A277795   |    |
%e A277795   o  o-o-o
%e A277795 (2)
%e A277795   o-o-o
%e A277795     |
%e A277795     o-o-o-o
%e A277795     |
%e A277795   o-o-o
%e A277795 (3)
%e A277795   o-o-o-o-o-o-o
%e A277795         |
%e A277795       o-o-o
%e A277795 (End)
%o A277795 (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
%Y A277795 Cf. A000055, A130131.
%K A277795 nonn
%O A277795 0,5
%A A277795 _Gabriel Burns_, Oct 31 2016
%E A277795 Corrections and more terms from _Andrey Zabolotskiy_, Nov 21 2016
%E A277795 a(24) onwards from _Andrew Howroyd_, Feb 06 2025