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
Keywords
Examples
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.
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.
Programs
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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);
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
Comments