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%I #52 Oct 16 2021 15:17:56
%S 1,3,11,44,184,789,3435,15100,66806,296870,1323318,5911972,26455294,
%T 118528793,531540891,2385375732,10710619014,48112492938,216195753066,
%U 971744791032,4368674392104,19643610378738,88339070102046,397313118498744,1787115246076764
%N Number of stacked directed animals on the triangular lattice.
%C Closely related to directed animals. A square lattice version exists.
%H Vincenzo Librandi, <a href="/A059714/b059714.txt">Table of n, a(n) for n = 1..200</a>
%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://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 M. Bousquet-Mélou and S. Butler, <a href="http://arXiv.org/abs/math.CO/0603617">Forest-like permutations</a>, arXiv:math/0603617 [math.CO], 2006.
%H Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%F G.f.: ((1-3*x)*(1-4*x)-(1-5*x)*sqrt(1-4*x))/(2*x*(2-9*x)).
%F 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.]
%F a(n) ~ 3^(2*n-1)/2^(n+2). - _Vaclav Kotesovec_, Oct 11 2012
%F 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
%F 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
%F 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
%F 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
%p 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:
%t 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 *)
%t 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 *)
%o (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
%o (Maxima)
%o 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 */
%Y Cf. A005773.
%K nonn
%O 1,2
%A _Mireille Bousquet-Mélou_, Feb 08 2001
%E More terms from _James A. Sellers_, Feb 09 2001
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