A121583 -id:A121583 - cp's OEIS frontend

A121581 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). 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, 1, 1, 1, 3, 2, 1, 9, 11, 3, 1, 33, 43, 39, 4, 1, 153, 193, 199, 169, 5, 1, 873, 1057, 1099, 1081, 923, 6, 1, 5913, 6937, 7147, 7171, 7027, 6117, 7, 1, 46233, 53017, 54187, 54403, 54307, 53413, 47311, 8, 1, 409113, 461257, 468907, 470203, 470323, 469483, 463399

Offset: 1

Emeric Deutsch, Aug 11 2006

Row sums are the factorials (A000142). T(n,0)=1; Sum(k*T(n,k), k=0..n)=A121582

			T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
Triangle starts:
1;
1,1;
1,3,2;
1,9,11,3;
1,33,43,39,4;

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

Cf. A000142, A121582, A100822, A121583.

Q[1]:=t: for n from 2 to 11 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))): P[1]:=1: P[n]:=subs(t=1,Q[n]): od: for n from 1 to 11 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form

The generating polynomial of row n is P(n,s)=Q(n,1,s), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

A121584 Number of cells in columns 1 and 2 of 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.

Original entry on oeis.org

1, 4, 18, 93, 569, 4074, 33336, 306035, 3111771, 34708944, 421407314, 5533007841, 78125977725, 1180594364966, 19012215609564, 325058642549919, 5880810783960431, 112243265407073100, 2254038189505807926

Offset: 1

Emeric Deutsch, Aug 11 2006

nonn

a(n)=Sum(A121583(n,k),k=1..2n-2) for n>=2. a(n)=A121580(n)+A121582(n)

			a(2)=4 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having a total of 2 cells in their first two columns.

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

Cf. A121580, A121582, A121583.

a[1]:=1: a[2]:=4: for n from 3 to 22 do a[n]:=((2*n-3)*a[n-1]-(n-1)*a[n-2])/(n-2)+(1/2)*(2*n^3-9*n^2+17*n-16)*(n-1)!/(n-2) od: seq(a[n],n=1..22);

a(1)=1, a(2)=4, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)]/(n-2) + (1/2)(2n^3-9n^2+17n-16)(n-1)!/(n-2) for n>=3.

Conjecture D-finite with recurrence 14*(-n+1)*a(n) +(14*n^2+1731*n-6995)*a(n-1) +3*(-577*n^2+480*n+7243)*a(n-2) +2*(2781*n^2-11952*n+6004)*a(n-3) +(-5987*n^2+36181*n-54220)*a(n-4) +2*(1071*n-3433)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A121581 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). 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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