OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana and Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.
FORMULA
G.f.: F(x) satisfies F(x) = H(F(x)) + x + (F(x)^2 + F(x^2))/2 where H(x)/x^2 is the g.f. of A257887.
EXAMPLE
For n=4, there are 11 planar tanglegrams of size 4.
PROG
(PARI) \\ here H(n)/x^2 is g.f. of A257887.
H(n)={(x - x^2 - serreverse(sum(k=0, n+1, (binomial(2*k, k)/(k+1))^2*x^(k+1)) + O(x^(n+3))))/2}
seq(n)={my(h=H(n-2), p=O(x)); for(n=1, n, p = subst(h + O(x*x^n), x, p) + x + (p^2 + subst(p, x, x^2))/2); Vec(p)} \\ Andrew Howroyd, Nov 18 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin Liu, Nov 16 2021
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Nov 18 2021
STATUS
approved