A059714 Number of stacked directed animals on the triangular lattice.

1, 3, 11, 44, 184, 789, 3435, 15100, 66806, 296870, 1323318, 5911972, 26455294, 118528793, 531540891, 2385375732, 10710619014, 48112492938, 216195753066, 971744791032, 4368674392104, 19643610378738, 88339070102046, 397313118498744, 1787115246076764

Offset

1,2

Comments

Closely related to directed animals. A square lattice version exists.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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.

M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Formula

G.f.: ((1-3*x)*(1-4*x)-(1-5*x)*sqrt(1-4*x))/(2*x*(2-9*x)).

2*(n+1)*a(n) +(5-27*n)*a(n-1) +(121*n-163)*a(n-2) +90*(5-2*n)*a(n-3) =0. - R. J. Mathar, Aug 14 2012 [See following Israel's contribution.]

a(n) ~ 3^(2*n-1)/2^(n+2). - Vaclav Kotesovec, Oct 11 2012

a(n) = 3^(2*n-2)/2^n*(2-Sum_{k=1..n-1} (k+8)*C(2*k,k)*2^k/((k+1)*(k+2)*3^(2*k)) ), for n>1. - Vaclav Kotesovec, Oct 28 2012

a(n) = Sum_{k=1..n-1} (k+1)*2^(k-1)*binomial(2*n,n-k-1)/n + binomial(2*n,n-1)/n. - Vladimir Kruchinin, Jun 08 2016

G.f. satisfies 60*x^3-31*x^2+4*x+(90*x^3-79*x^2+22*x-2)*g(x)+(180*x^4-121*x^3+27*x^2-2*x)*g'(x) = 0, from which Mathar's recurrence follows. - Robert Israel, Jun 08 2016

G.f. F satisfies 0 = F^2*(9*x^2 - 2*x) + F*(12*x^2 - 7*x + 1) + 4*x^2 - x. - F. Chapoton, Oct 16 2021

Maple

gf := ((1-3*x)*(1-4*x)-(1-5*x)*sqrt(1-4*x))/(2*x*(2-9*x)): s := series(gf, x, 50): for i from 1 to 100 do printf(`%d, `, coeff(s, x, i)) od:

Mathematica

Rest[Table[SeriesCoefficient[((1-3*x)*(1-4*x)-(1-5*x)*Sqrt[1-4*x])/(2*x*(2-9*x)), {x, 0, n}], {n, 0, 20}]] (* Vaclav Kotesovec, Oct 28 2012 *)
Flatten[{1, Table[3^(2*n-2)/2^n* (2 - Sum[(k+8)*Binomial[2*k, k]*2^k/((k+1)*(k+2)*3^(2*k)), {k, 1, n-1}]), {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

Prog

(PARI) x = 'x + O('x^40); Vec(((1-3*x)*(1-4*x)-(1-5*x)*sqrt(1-4*x))/(2*x*(2-9*x))) \\ Michel Marcus, Jan 28 2016
(Maxima)
a(n):=sum((k+1)*2^(k-1)*binomial(2*n, n-k-1), k, 1, n-1)/n+binomial(2*n, n-1)/n; /* Vladimir Kruchinin, Jun 08 2016 */

Crossrefs

Cf. A005773.

Sequence in context: A149073 A167011 A319322 * A059715 A026748 A113174

Adjacent sequences: A059711 A059712 A059713 * A059715 A059716 A059717

Keyword

nonn

Author

Mireille Bousquet-Mélou, Feb 08 2001

Extensions

More terms from James A. Sellers, Feb 09 2001

Status

approved