A059712 - OEIS

%I #26 Jun 28 2024 12:10:10

%S 1,2,6,19,63,213,729,2513,8703,30232,105236,366849,1280131,4470354,

%T 15619386,54595869,190891131,667590414,2335121082,8168950665,

%U 28580354769,100000811433,349918126509,1224476796543,4285005630969

%N Number of stacked directed animals on the square lattice.

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

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="https://www.labri.fr/Perso/~bousquet/Articles/Klarner/article.ps.gz">Lattice animals and heaps of dimers</a>

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00352-7">Lattice animals and heaps of dimers</a>, Discrete Math. 258 (2002), no. 1-3, 235-274.

%H Florian Schager and Michael Wallner, <a href="https://arxiv.org/abs/2406.16417">A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes</a>, arXiv:2406.16417 [math.CO], 2024. See p. 5.

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

%F 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

%F 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

%F BINOMIAL transform is A059714. HANKEL transform is A001519(n+1). - _Michael Somos_, Apr 17 2012

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

%p 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:

%t 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 *)

%o (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 */

%Y Directed animals: A005773.

%Y Cf. A001519, A059714.

%K nonn

%O 1,2

%A _Mireille Bousquet-Mélou_, Feb 08 2001

%E More terms from _James A. Sellers_, Feb 09 2001