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A181368 Triangle read by rows: T(n,k) is the number of L-convex polyominoes of semiperimeter n, having k maximal rectangles (n >= 2, 1 <= k <= floor(n/2)). An L-convex polyomino is a convex polyomino in which any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientations of the letter L). A maximal rectangle in an L-convex polyomino P is a rectangle included in P that is maximal with respect to inclusion.

%I A181368 #8 Nov 03 2019 19:40:53
%S A181368 1,2,3,4,4,20,5,61,16,6,146,128,7,301,584,64,8,560,1992,704,9,966,
%T A181368 5641,4272,256,10,1572,14002,18880,3584,11,2442,31471,67820,27136,
%U A181368 1024,12,3652,65428,209820,147200,17408,13,5291,127699,579125,640096,157952
%N A181368 Triangle read by rows: T(n,k) is the number of L-convex polyominoes of semiperimeter n, having k maximal rectangles (n >= 2, 1 <= k <= floor(n/2)). An L-convex polyomino is a convex polyomino in which any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientations of the letter L). A maximal rectangle in an L-convex polyomino P is a rectangle included in P that is maximal with respect to inclusion.
%C A181368 Row n contains floor(n/2) entries.
%C A181368 Sum of entries in row n is A003480(n-2).
%C A181368 Sum_{k>=1} k*T(n,k) = A181369(n).
%D A181368 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741 (see Fig. 9).
%D A181368 G. Castiglione and A. Restivo, Reconstruction of L-convex polyominoes, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003.
%F A181368 T(n+2,k+1) = Sum_{j=0..2k} (-1)^j*2^(2k-j)*binomial(2k, j)*binomial(n+2k-j+1, 4k+1).
%F A181368 T(n+2,k+1) = Sum_{j=0..2k} binomial(2k, j)*binomial(n+j+1, 2k+j+1).
%F A181368 G.f. = G(t,z) = t*z^2*(1-z)^2/((1-z)^4 - t*z^2*(2-z)^2).
%e A181368 T(3,1)=2 because the L-convex polyominoes of semiperimeter 3 are the horizontal and the vertical dominoes, each containing one maximal rectangle.
%e A181368 Triangle starts:
%e A181368   1;
%e A181368   2;
%e A181368   3,   4;
%e A181368   4,  20;
%e A181368   5,  61,  16;
%e A181368   6, 146, 128;
%p A181368 T := proc (n, k) options operator, arrow: sum(binomial(2*k-2, j)*binomial(n+j-1, 2*k+j-1), j = 0 .. 2*k-2) end proc: for n from 2 to 14 do seq(T(n, k), k = 1 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A181368 Cf. A003480, A181369.
%K A181368 nonn,tabf
%O A181368 2,2
%A A181368 _Emeric Deutsch_, Oct 17 2010