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

Author

Mireille Bousquet-Mélou, Feb 08 2001

Keywords

• nonn

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.

Crossrefs

Cf. A005773.

Programs

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 *)

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 */

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

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

Extensions

More terms from James Sellers, Feb 09 2001