OFFSET
1,14
COMMENTS
The underlying tree structure before the weights are applied (Z with weight one, U with weight three) is a series-reduced tree because a non-leaf node Z has at least two children.
REFERENCES
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.
LINKS
Marko Riedel et al., Math.StackExchange, Counting plane trees
FORMULA
G.f.: (1 + z^4 - sqrt(z^8 - 4*z^5 - 2*z^4 +1))/(2*(z+1)).
a(n) = Sum_{k=floor(n/5)+1..floor((n-1)/4)} (1/(n-3*k)) * binomial(n-3*k,k) * binomial(k-2, n-4*k-1)) for n >= 1, n <> 4. a(4) = 1.
D-finite with recurrence: n*a(n) +(n)*a(n-1) +(n-2)*a(n-2) +(n-2)*a(n-3) +2*(-n+6)*a(n-4) +6*(-n+7)*a(n-5) +2*(-3*n+23)*a(n-6) +6*(-n+9)*a(n-7) +(-3*n+26)*a(n-8) +(n-12)*a(n-9) +(n-14)*a(n-10) +(n-14)*a(n-11)=0. - R. J. Mathar, Jan 27 2020
EXAMPLE
For n=4, the tree is Z-U, for n=9 the tree is
Z-U
/
Z
\
Z-U.
CROSSREFS
KEYWORD
nonn
AUTHOR
Marko Riedel, Jan 09 2020
STATUS
approved