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

2,2

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).

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.

Links

Table of n, a(n) for n=2..49.

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).

Example

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;

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

Crossrefs

Cf. A003480, A181369.

Sequence in context: A185417 A214384 A118022 * A037848 A037884 A347728

Adjacent sequences: A181365 A181366 A181367 * A181369 A181370 A181371

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 17 2010

Status

approved