OFFSET
1,4
COMMENTS
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.
REFERENCES
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]
LINKS
FORMULA
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.
MAPLE
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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Nov 21 2004
Entry revised by N. J. A. Sloane, Feb 26 2007
STATUS
approved