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%I #8 Aug 04 2022 15:54:52
%S 0,2,6,60,500,7290,100842,1978368,38263752,949218750,23579476910,
%T 709026379776,21505924728444,760772509715764,27246730957031250,
%U 1109165339867873280,45798768824157052688,2109518949433090534902
%N Number of labeled rooted trees on [n] that have a primary branch.
%C This sequence is the labeled version of A027415 where the definition can be found.
%H N. J. A. Sloane, <a href="/A000081/a000081.html">Illustration of initial terms</a>
%F a(n) = Sum_{i=1..floor(n/2)} binomial(n,n-i)*r(n-i)*r(i) where r(i) = A000169(i).
%F a(n) = A000169(n) - A356073(n).
%e a(5) = 500. In the illustrations by Sloane found in the link above, for n = 5, there are A027415(5) = 6 rooted trees with a primary branch: the first six trees shown.
%t Table[Sum[Binomial[n, n - i] (n - i)^(n - i - 1)*i^(i - 1), {i, 1,Floor[n/2]}], {n, 1, 20}]
%Y Cf. A027415, A356074, A000169.
%K nonn
%O 1,2
%A _Geoffrey Critzer_, Jul 31 2022
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