A121754 Number of columns ending at an even level in all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

0, 1, 6, 31, 211, 1530, 13086, 120888, 1260792, 14140080, 174692880, 2304970560, 32969263680, 500368821120, 8139251433600, 139686867532800, 2547638477798400, 48786683184691200, 986263089841612800

Offset

1,3

Comments

a(n)=Sum(k*A121698(n,k),k=1..n-1).

References

E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Links

Table of n, a(n) for n=1..19.

Formula

Recurrence relation: a(n)=(2n-3)a(n-1)-(n-1)(n-3)a(n-2)+(n-2)![n-2+(1/2)(1+(-1)^(n-1))(n-1)] for n>=3; a(1)=0, a(2)=1.

Conjecture D-finite with recurrence 16*(n+1)*a(n) +(-16*n^2-178*n+531)*a(n-1) +(-16*n^3+178*n^2-393*n-510)*a(n-2) +(16*n^4+98*n^3-1439*n^2+4222*n-3623)*a(n-3) +(-146*n^4+1479*n^3-4483*n^2+3054*n+2841)*a(n-4) +(130*n-311)*(n-6)*(-4+n)^2*a(n-5)=0. - R. J. Mathar, Jul 26 2022

Example

a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 1 and 0 columns ending at an even level, respectively.

Maple

a[1]:=0: a[2]:=1: for n from 3 to 22 do a[n]:=(2*n-3)*a[n-1]-(n-1)*(n-3)*a[n-2]+(n-2)!*(n-2+(1/2)*(1+(-1)^(n-1))*(n-1)) od: seq(a[n], n=1..22);

Crossrefs

Cf. A121698, A121752.

Sequence in context: A361306 A368535 A362387 * A351818 A200775 A244255

Adjacent sequences: A121751 A121752 A121753 * A121755 A121756 A121757

Keyword

nonn

Author

Emeric Deutsch, Aug 23 2006

Status

approved