A059715 Number of multi-directed animals on the triangular lattice.

1, 3, 11, 44, 184, 790, 3450, 15242, 67895, 304267, 1369761, 6188002, 28031111, 127253141, 578694237, 2635356807, 12015117401, 54831125131, 250418753498, 1144434017309

Offset: 1

Author

Mireille Bousquet-Mélou, Feb 08 2001

Keywords

• nonn
• more

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.

Crossrefs

Cf. A000108, A005773.

Programs

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 *)

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