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A128611 Number of Z-convex polyominoes with semiperimeter n.

%I A128611 #21 Nov 20 2022 05:29:22
%S A128611 0,0,1,2,7,28,116,484,2022,8448,35290,147376,615228,2567060,10704976,
%T A128611 44611804,185780308,773060804,3214225836,13352979316,55426067494,
%U A128611 229870371888,952548347122,3943943111920,16316243701350,67447113649312,278592165886198,1149863118820584,4742473257979906,19545876370622104,80502059920697442
%N A128611 Number of Z-convex polyominoes with semiperimeter n.
%H A128611 Robert Israel, <a href="/A128611/b128611.txt">Table of n, a(n) for n = 0..1658</a>
%H A128611 Adrien Boussicault, Simone Rinaldi, and Samanta Socci, <a href="https://arxiv.org/abs/1501.00872">The number of directed k-convex polyominoes</a>, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See page 2.
%H A128611 E. Duchi, S. Rinaldi and G. Schaeffer, <a href="https://arxiv.org/abs/math/0602124">The number of Z-convex polyominoes</a>, arXiv:math/0602124 [math.CO], 2006.
%F A128611 The Duchi paper has a g.f.
%F A128611 Asymptotically, a(n) ~ n/24 * 4^n.
%F A128611 G.f.: Let d:=(1-2*t-sqrt(1-4*t))/2; then g.f. is 2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) ) + t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ). - _N. J. A. Sloane_, Oct 02 2011
%F A128611 (-960+384*n)*a(n)+(1760-992*n)*a(n+1)+(-924+984*n)*a(2+n)+(64-490*n)*a(n+3)+(82+131*n)*a(n+4)+(-24-18*n)*a(n+5)+(2+n)*a(n+6), a(0) = 0. - _Robert Israel_, Aug 17 2018
%p A128611 d:=(1-2*t-sqrt(1-4*t))/2:
%p A128611 t1:=
%p A128611 2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) )
%p A128611 + t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ):
%p A128611 series(t1,t,120):
%p A128611 seriestolist(%); # _N. J. A. Sloane_, Oct 02 2011
%t A128611 gf = 2 t^4 (1-2t)^2 d/((1-4t)^2 (1-3t)(1-t)) + t^2 (1-6t+10t^2-2t^3-t^4)/ ((1-4t)(1-3t)(1-t)) /. d -> (1-2t-Sqrt[1-4t])/2;
%t A128611 CoefficientList[gf + O[t]^31, t] (* _Jean-François Alcover_, Aug 17 2018 *)
%Y A128611 Cf. A003480, A005436.
%K A128611 nonn
%O A128611 0,4
%A A128611 _Ralf Stephan_, May 08 2007