A121750 Number of columns of even length 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, 5, 26, 184, 1338, 11652, 108210, 1140336, 12849714, 159858900, 2117522754, 30442090248, 463511103426, 7569181895436, 130254363597330, 2383020441932256, 45738553437874962, 927010880040945924

Offset

1,3

Links

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

Elena Barcucci, Sara Brunetti, and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.

Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Formula

a(n) = Sum_{k=0..n-1} k*A121748(n,k).

Recurrence relation: a(n) = n*a(n-1)+d(n-1)+(n-1)!*floor((n-1)/2) for n>=2, a(1)=0, where d(1)=1, d(2)=0, d(2n)=3!+5!+...+(2n-1)!, d(2n+1)=-d(2n).

Conjecture D-finite with recurrence a(n) +(-n-2)*a(n-1) +(-n^2+4*n-2)*a(n-2) +(n^3-3*n^2-4*n+11)*a(n-3) -(n-1)*(n^2-11*n+26)*a(n-4) +(-n^3+5*n^2+7*n-43)*a(n-5) +(n-3)*(n-5)^2*a(n-6)=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 of even length, respectively.

Maple

d:=proc(n) if n=1 then 1 elif n=2 then 0 elif n mod 2 = 0 then add((2*j-1)!, j=2..n/2) else -d(n-1) fi end: a[1]:=0: for n from 2 to 22 do a[n]:=n*a[n-1]+d(n-1)+(n-1)!*floor((n-1)/2) od: seq(a[n], n=1..22);

Mathematica

d[n_] := Which[n == 1, 1, n == 2, 0, EvenQ[n], Sum[(2j - 1)!, {j, 2, n/2}], True, -d[n-1]];
a[n_] := a[n] = If[n == 1, 0, n*a[n-1] + d[n-1] + (n-1)!*Floor[(n-1)/2]];
Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Aug 20 2024, after Maple program *)

Crossrefs

Cf. A121747, A121748.

Sequence in context: A302896 A368176 A175151 * A143341 A303970 A366215

Adjacent sequences: A121747 A121748 A121749 * A121751 A121752 A121753

Keyword

nonn

Author

Emeric Deutsch, Aug 20 2006

Status

approved