%I M4638 #44 Jun 04 2024 17:08:00
%S 1,9,55,286,1362,6143,26729,113471,473471,1951612,7974660,32384127,
%T 130926391,527657073,2121795391,8518575466,34162154550,136893468863,
%U 548253828965,2194897467395,8784784672511,35153438973304,140653028240520,562719731644671
%N Number of convex polygons of length 2n on square lattice whose leftmost bottom vertex and rightmost top vertex have the same x-coordinate.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H M.-P. Delest and G. Viennot, <a href="https://doi.org/10.1016/0304-3975(84)90116-6">Algebraic languages and polyominoes enumeration</a>, Theoretical Computer Sci., 34 (1984), 169-206.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-55,120,-125,54,-8).
%F a(n) = A005436(n) - A005768(n) - A005769(n).
%F G.f.: x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003
%F a(n) = 12*a(n-1) - 55*a(n-2) + 120*a(n-3) - 125*a(n-4) + 54*a(n-5) - 8*a(n-6) for n > 8. - _Stefano Spezia_, Jun 04 2024
%p A005770:=(1-3*z+2*z**2+z**3)/(4*z-1)/(2*z-1)/(z**2-3*z+1)**2; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)),{x,0,28}],x] (* _Stefano Spezia_, Jun 04 2024 *)
%Y Cf. A005436, A005768, A005769.
%K nonn,easy
%O 5,2
%A _Simon Plouffe_, _N. J. A. Sloane_
%E Better description from Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003
%E More terms from _Sean A. Irvine_, Aug 26 2016