OFFSET
1,2
COMMENTS
Counts certain animals that generalize directed animals. They are also equinumerous with a class of n-ominoes studied by Klarner in 1967.
LINKS
M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers
M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Math. 258 (2002), no. 1-3, 235-274.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014.
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
FORMULA
The generating function is known in closed form. It is big and non-D-finite.
Bultel-Giraudo (2014), Prop. 3.2, give a g.f. - N. J. A. Sloane, Sep 21 2014
Conjecture: a(n) = Sum_{j=0..n-1} R(n-1, j) for n > 0 where R(n, j) = Sum_{p=0..n - j - 1} binomial(j + p + 2, p + 1)*R(n - j - 1, p) for 0 <= j < n with R(n, n) = 1. - Mikhail Kurkov, Aug 09 2023
a(n) ~ c * d^n, where d = 4.5878943629412631496341355193804435266001072071... and c = 0.0653089423402623226212483954648487116904937... - Vaclav Kotesovec, Aug 13 2023
MATHEMATICA
terms = 12;
c[g_, t_] := c[g, t] = Sum[c[g, n, t], {n, 0, 2 terms}];
c[g_, n_, t_] := c[g, n, t] = P[g, n, t] - Sum[c[g, k, t] P[g, n-k-1, t], {k, 0, n-1}];
P[g_, n_, t_] := 1/F[g, n, t];
F[g_, n_, t_] := F[g, n, t] = If[n<=g, 1, F[g, n-1, t] - t F[g, n-g-1, t]];
Rest[CoefficientList[1-1/c[1, t] + O[t]^(terms+1), t]][[1 ;; terms]] (* Jean-François Alcover, Jul 25 2018 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Mireille Bousquet-Mélou, Feb 08 2001
STATUS
approved