A059712 Number of stacked directed animals on the square lattice.

1, 2, 6, 19, 63, 213, 729, 2513, 8703, 30232, 105236, 366849, 1280131, 4470354, 15619386, 54595869, 190891131, 667590414, 2335121082, 8168950665, 28580354769, 100000811433, 349918126509, 1224476796543, 4285005630969

Offset: 1

Author

Mireille Bousquet-Mélou, Feb 08 2001

Keywords

• nonn

Comments

The generating function is simply derived from the generating function for directed animals. A triangular lattice version exists.

Examples

			x + 2*x^2 + 6*x^3 + 19*x^4 + 63*x^5 + 213*x^6 + 729*x^7 + ...
		

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.
• Florian Schager and Michael Wallner, A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes, arXiv:2406.16417 [math.CO], 2024. See p. 5.

Crossrefs

Directed animals: A005773.

Cf. A001519, A059714.

Programs

Maple

gf := ((1-2*x)*(1-3*x)-(1-4*x)*sqrt((1-3*x)*(1+x)))/(2*x*(2-7*x)): s := series(gf, x, 50): for i from 1 to 100 do printf(`%d,`,coeff(s,x,i)) od:

Mathematica

CoefficientList[ ((1-2*x)*(1-3*x)-(1-4*x)*Sqrt[(1-3*x)*(1+x)])/(2*x*(2-7*x)) + O[x]^30, x] // Rest (* Jean-François Alcover, Jun 19 2015 *)

PARI

{a(n) = local(A); if( n<1, 0, A = O(x); for( k=1, ceil(n/2), A = 1/( 1/x - 2 - (2 - 7*x) / (1 - 3*x) * A)); polcoeff(A, n))} /* Michael Somos, Apr 17 2012 */

Formula

G.f.: ((1-2x)(1-3x)-(1-4x)sqrt((1-3x)(1+x)))/(2x(2-7x)).

G.f. A(x) satisfies 0 = f(x, A(x)) where f(x ,y) = (7*x^2 - 2*x) * y^2 + (6*x^2 - 5*x + 1) * y + (3*x^2 - x). - Michael Somos, Apr 17 2012

0 = (105*n^2 + 861*n) * a(n) + (40*n^2 + 433*n + 672) * a(n+1) - (55*n^2 + 586*n + 1200) * a(n+2) + (10*n^2 + 112*n + 288) * a(n+3). - Michael Somos, Apr 17 2012

BINOMIAL transform is A059714. HANKEL transform is A001519(n+1). - Michael Somos, Apr 17 2012

Extensions

More terms from James Sellers, Feb 09 2001