A121751 -id:A121751 - cp's OEIS frontend

A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (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, 2, 2, 1, 1, 4, 8, 7, 3, 2, 14, 32, 37, 23, 10, 4, 44, 142, 207, 180, 97, 38, 12, 194, 730, 1267, 1327, 911, 425, 150, 36, 812, 3810, 8104, 10387, 8876, 5257, 2222, 708, 144, 4362, 23284, 56987, 84792, 85317, 60814, 31368, 11972, 3408, 576, 22716

Offset: 0

Emeric Deutsch, Aug 23 2006

Row sums are the factorials (A000142).
T(n,0) = A121751(n), T(n,n) = A010551(n-1) for n>=1.
Sum_{k=0..n} k*T(n,k) = A121752(n).

			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 ending at an odd level, respectively.
Triangle starts:
  1;
  0,1;
  1,0,1;
  2,2,1,1;
  4,8,7,3,2;
  14,32,37,23,10,4;

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.

Cf. A000142, A010551, A121751, A121752, A121698.

Q[0]:=1: Q[1]:=t: for n from 2 to 10 do Q[n]:=expand(subs({t=s,s=t},Q[n-1])+(t*floor(n/2)+s*floor((n-1)/2))*Q[n-1]) od: for n from 0 to 10 do P[n]:=sort(subs(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

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

A121753 Number of deco polyominoes of height n in which all columns end at an odd level. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Original entry on oeis.org

1, 1, 2, 6, 16, 62, 230, 1114, 5268, 30702, 176226, 1201638, 8107464, 63339702, 491010102, 4324845834, 37867131900, 371275954758, 3623124865986, 39137296073094, 421150512316032, 4969568447400366, 58455531552960198

Offset: 1

Emeric Deutsch, Aug 23 2006

nonn

a(n)=A121698(n,0).

			a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes and only the horizontal one has all of its columns ending at an odd level.

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.

Cf. A121698, A121751.

a[1]:=1: a[2]:=1: for n from 3 to 26 do a[n]:= (1+2*floor((n-2)/2))*a[n-1]-(floor((n-1)/2)*floor((n-2)/2)-1)*a[n-2] od: seq(a[n],n=1..26);

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

cp's OEIS Frontend

A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (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.

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