%I #16 Aug 09 2017 02:48:35
%S 0,1,1,3,6,17,37,102,239,658,1607,4425,11185,30990,80070,222731,
%T 586218,1638333,4370721,12262003,33077327,93128828,253454781,
%U 715784848,1962537755,5557799401,15332668869,43527249088,120716987723
%N Number of rooted unlabeled trees on n nodes having a primary branch.
%C Let T be a tree with root node R. If R and the edges incident with it are deleted, the resulting rooted trees are called branches. A primary branch (there can be at most one) has i nodes where n/2 <= i <= n-1.
%D A. Meir and J. W. Moon, On the branch-sizes of rooted unlabeled trees, in "Graph Theory and Its Applications", Annals New York Acad. Sci., Vol. 576, 1989, pp. 399-407. [MR 1110839]
%H N. J. A. Sloane, <a href="/A027415/b027415.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Let r(n) = A000081(n) = number of rooted trees on n nodes. Then a(n)=sum(r(n-i)*r(i), i=1..floor(n/2)) - _Emeric Deutsch_, Nov 21 2004. Comment from _N. J. A. Sloane_: The term r(n-i) gives the number of ways of picking the primary branch, while the term r(i) gives the number of ways of picking the rest of the tree including the root R.
%p N := 50: Y := [ 1,1 ]: for n from 3 to N do x*mul( (1-x^i)^(-Y[ i ]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); Y := [ op(Y),b ]; od: P:=n->sum(Y[n-i]*Y[i],i=1..floor(n/2)): seq(P(n),n=1..35); # _Emeric Deutsch_, Nov 21 2004
%Y This sequence + A027416 = A000081. Cf. A000081, A000055, A102911.
%K nonn
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Nov 21 2004
%E Entry revised by _N. J. A. Sloane_, Feb 26 2007