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.

1, 2, 3, 4, 4, 20, 5, 61, 16, 6, 146, 128, 7, 301, 584, 64, 8, 560, 1992, 704, 9, 966, 5641, 4272, 256, 10, 1572, 14002, 18880, 3584, 11, 2442, 31471, 67820, 27136, 1024, 12, 3652, 65428, 209820, 147200, 17408, 13, 5291, 127699, 579125, 640096, 157952

Offset: 2

Author

Emeric Deutsch, Oct 17 2010

Keywords

• nonn
• tabf

Comments

Row n contains floor(n/2) entries.

Sum of entries in row n is A003480(n-2).

Sum_{k>=1} k*T(n,k) = A181369(n).

Examples

			T(3,1)=2 because the L-convex polyominoes of semiperimeter 3 are the horizontal and the vertical dominoes, each containing one maximal rectangle.
Triangle starts:
  1;
  2;
  3,   4;
  4,  20;
  5,  61,  16;
  6, 146, 128;
		

References

• 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).
• G. Castiglione and A. Restivo, Reconstruction of L-convex polyominoes, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003.

Crossrefs

Cf. A003480, A181369.

Programs

Maple

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

Formula

T(n+2,k+1) = Sum_{j=0..2k} (-1)^j*2^(2k-j)*binomial(2k, j)*binomial(n+2k-j+1, 4k+1).

T(n+2,k+1) = Sum_{j=0..2k} binomial(2k, j)*binomial(n+j+1, 2k+j+1).

G.f. = G(t,z) = t*z^2*(1-z)^2/((1-z)^4 - t*z^2*(2-z)^2).