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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A336206

Adjacent sequences:  A181365 A181366 A181367 * A181369 A181370 A181371

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Oct 17 2010

STATUS

approved

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Last modified August 10 04:38 EDT 2020. Contains 336368 sequences. (Running on oeis4.)