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A121302 Number of directed column-convex polyominoes having at least one 1-cell column.

%I A121302 #9 Jul 22 2022 08:47:59
%S A121302 1,1,4,10,28,75,202,540,1440,3828,10153,26875,71021,187421,494013,
%T A121302 1300844,3422509,8998118,23642479,62088032,162978242,427648023,
%U A121302 1121766397,2941697012,7712415568,20215976824,52981414253,138831400836
%N A121302 Number of directed column-convex polyominoes having at least one 1-cell column.
%C A121302 a(n) = Fibonacci(2n-1) - A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n having k 1-cell columns). Column 1 of A121301.
%H A121302 E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%H A121302 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-2,4,-1).
%F A121302 G.f.: z(1-z)(1-3z+2z^2)/[(1-3z+z^2)(1-2z-z^2+z^3)].
%F A121302 a(n) = A001519(n)-A077998(n-2), n>0. - _R. J. Mathar_, Jul 22 2022
%e A121302 a(3)=4 because, with the exception of the 3-cell column, all the other four directed column-convex polyominoes of area 3 have a 1-cell column.
%p A121302 G:=z*(1-z)*(1-3*z+2*z^2)/(1-3*z+z^2)/(1-2*z-z^2+z^3): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=1..32);
%o A121302 (PARI) Vec(z*(1-z)*(1-3*z+2*z^2)/((1-3*z+z^2)*(1-2*z-z^2+z^3)) + O(z^40)) \\ _Michel Marcus_, Feb 14 2016
%Y A121302 Cf. A121469, A121301.
%K A121302 nonn,easy
%O A121302 1,3
%A A121302 _Emeric Deutsch_, Aug 04 2006