A121745 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of odd length (0<=k<=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.

1, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 9, 4, 2, 9, 28, 36, 30, 13, 4, 27, 107, 195, 198, 130, 51, 12, 117, 532, 1106, 1371, 1096, 585, 197, 36, 459, 2573, 6547, 9914, 9886, 6733, 3132, 932, 144, 2421, 15139, 43606, 76017, 88793, 72689, 42200, 17027, 4412, 576, 11979, 87362, 293146, 597856, 824343, 807930, 574837, 296296, 107231, 24940, 2880

Offset

0,8

Comments

Row sums are the factorials (A000142).

T(n,0) = A121746(n).

Sum_{k=0..n} k * T(n,k) = A121747(n).

Links

Table of n, a(n) for n=0..65.

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

Formula

The row generating polynomials P[n](t) are given by P[n](t)=Q[n](t,1,t,1), where Q[n](t,s,x,y) are defined by Q[n](t,s,x,y)=Q[n-1](t,s,y,x)+[floor(n/2)*x+floor((n-1)/2)*y]Q[n-1](t,s,t,s) for n>=2 and Q[0](t,s,x,y)=1, Q[1](t,s,x,y]=x.

Example

T(2,0)=1, T(2,1)=0 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns of odd length, respectively.
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  1,  3,  1,  1;
  3,  6,  9,  4,  2;
  9, 28, 36, 30, 13, 4;
  ...

Maple

Q[0]:=1: Q[1]:=x: for n from 2 to 10 do Q[n]:=expand(subs({x=y, y=x}, Q[n-1])+(floor(n/2)*x+floor((n-1)/2)*y)*subs({x=t, y=s}, Q[n-1])) od: for n from 0 to 10 do P[n]:=sort(subs({x=t, y=1, s=1}, Q[n])) od: for n from 0 to 10 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

Mathematica

Q[0] = 1; Q[1] = x;
Do[Q[n] = Expand[ReplaceAll[Q[n-1], {x -> y, y -> x}] + (Floor[n/2]*x + Floor[(n-1)/2]*y)*ReplaceAll[Q[n-1], {x -> t, y -> s}]] , {n, 2, 10}];
Do[P[n] = ReplaceAll[Q[n], {x -> t, y -> 1, s -> 1}], {n, 0, 10}];
Table[Coefficient[P[n], t, j], {n, 0, 10}, {j, 0, n}] // Flatten (* Jean-François Alcover, Aug 19 2024, after Maple program *)

Crossrefs

Cf. A000142, A121746, A121747, A121748.

Sequence in context: A156710 A114588 A253223 * A252983 A089312 A246674

Adjacent sequences: A121742 A121743 A121744 * A121746 A121747 A121748

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 20 2006

Status

approved