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A059712 Number of stacked directed animals on the square lattice. 0
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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Last modified October 28 10:27 EDT 2021. Contains 348327 sequences. (Running on oeis4.)