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

1,2

Comments

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

Links

Table of n, a(n) for n=1..25.

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Directed animals: A005773.

Cf. A001519, A059714.

Sequence in context: A191993 A120900 A284216 * A059713 A109262 A006724

Adjacent sequences: A059709 A059710 A059711 * A059713 A059714 A059715

Keyword

nonn

Author

Mireille Bousquet-Mélou, Feb 08 2001

Extensions

More terms from James A. Sellers, Feb 09 2001

Status

approved