cp's OEIS Frontend

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

%I A059715 #34 Apr 24 2024 08:09:20
%S A059715 1,3,11,44,184,790,3450,15242,67895,304267,1369761,6188002,28031111,
%T A059715 127253141,578694237,2635356807,12015117401,54831125131,250418753498,
%U A059715 1144434017309
%N A059715 Number of multi-directed animals on the triangular lattice.
%C A059715 Counts certain animals that generalize directed animals. They are also equinumerous with a class of n-ominoes studied by Klarner in 1967.
%H A059715 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://www.labri.fr/Perso/~bousquet/Articles/Klarner/article.ps.gz">Lattice animals and heaps of dimers</a>
%H A059715 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00352-7">Lattice animals and heaps of dimers</a>, Discrete Math. 258 (2002), no. 1-3, 235-274.
%H A059715 J.-P. Bultel and S. Giraudo, <a href="http://arxiv.org/abs/1406.6903">Combinatorial Hopf algebras from PROs</a>, arXiv preprint arXiv:1406.6903 [math.CO], 2014.
%H A059715 D. A. Klarner, <a href="http://dx.doi.org/10.4153/CJM-1967-080-4">Cell growth problems</a>, Canad. J. Math. 19 (1967) 851-863.
%F A059715 The generating function is known in closed form. It is big and non-D-finite.
%F A059715 Bultel-Giraudo (2014), Prop. 3.2, give a g.f. - _N. J. A. Sloane_, Sep 21 2014
%F A059715 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
%F A059715 a(n) ~ c * d^n, where d = 4.5878943629412631496341355193804435266001072071... and  c = 0.0653089423402623226212483954648487116904937... - _Vaclav Kotesovec_, Aug 13 2023
%t A059715 terms = 12;
%t A059715 c[g_, t_] := c[g, t] = Sum[c[g, n, t], {n, 0, 2 terms}];
%t A059715 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}];
%t A059715 P[g_, n_, t_] := 1/F[g, n, t];
%t A059715 F[g_, n_, t_] := F[g, n, t] = If[n<=g, 1, F[g, n-1, t] - t F[g, n-g-1, t]];
%t A059715 Rest[CoefficientList[1-1/c[1, t] + O[t]^(terms+1), t]][[1 ;; terms]] (* _Jean-François Alcover_, Jul 25 2018 *)
%Y A059715 Cf. A000108, A005773.
%K A059715 nonn,more
%O A059715 1,2
%A A059715 _Mireille Bousquet-Mélou_, Feb 08 2001