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A059714 Number of stacked directed animals on the triangular lattice. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

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 *)

PROG

(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

(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 */

CROSSREFS

Cf. A005773.

Sequence in context: A149073 A167011 A319322 * A059715 A026748 A113174

Adjacent sequences:  A059711 A059712 A059713 * A059715 A059716 A059717

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 November 28 13:47 EST 2021. Contains 349413 sequences. (Running on oeis4.)