OFFSET
1,5
COMMENTS
a(n) is the number of maximal intervals of balanced binary trees in the Tamari lattice of binary trees with n internal nodes. An interval of balanced binary trees in the Tamari lattice is maximal if it is not strictly included in another interval of balanced binary trees.
LINKS
Joerg Arndt, Table of n, a(n) for n = 1..1000
S. Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv preprint arXiv:1107.3472 [math.CO], 2011.
S. Giraudo, Intervals of balanced binary trees in the Tamari lattice, Theoretical Computer Science, 420, 1--27, 2012.
FORMULA
G.f.: A(x) = B(x, 0, 0, 0) where B(x, y, z, t) satisfies B(x, y, z, t) = x + B(x^2 + 2*y*t + y*z, x, x^2 + x*y, y*t + y*z).
MATHEMATICA
m = 47; R = O[x]^(m + 1);
B[x_, y_, z_, t_, k_:0] = If[k >= m, x, x + R + B[x^2 + 2 y t + y z + R, x + R, x^2 + x y + R, y t + y z + R, k + 1] // Normal];
CoefficientList[B[x, 0, 0, 0], x] // Rest (* Jean-François Alcover, Dec 16 2018, from Joerg Arndt's PARI code *)
PROG
(PARI) N = 66; R = O('x^(N+1)); x = 'x+R;
B(x, y, z, t, k=0) = if( k>=N, x, x + R + B(x^2 + 2*y*t + y*z + R, x + R, x^2 + x*y + R, y*t + y*z + R, k+1) );
Vec( B(x, 0, 0, 0) ) \\ Joerg Arndt, May 01 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Samuele Giraudo, Apr 28 2016
EXTENSIONS
Term a(33) and beyond from Joerg Arndt, May 01 2016
STATUS
approved