A121555 -id:A121555 - cp's OEIS frontend

A121554 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (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, 7, 7, 6, 3, 1, 30, 35, 30, 18, 6, 1, 157, 205, 184, 117, 46, 10, 1, 972, 1392, 1304, 874, 381, 101, 15, 1, 6961, 10764, 10499, 7355, 3470, 1052, 197, 21, 1, 56660, 93493, 94668, 68909, 34622, 11606, 2542, 351, 28, 1, 516901, 901900

Offset: 0

Emeric Deutsch, Aug 08 2006

Row sums are the factorials (A000142). T(n,0)=A001053(n). Sum(k*T(n,k), k=0..n)=A121555(n).

			T(2,0)=1, T(2,1)=0, T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
Triangle starts:
1;
0,1;
1,0,1;
2,2,1,1;
7,7,6,3,1;
30,35,30,18,6,1;

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

Cf. A000142, A001053, A121555.

Q[0]:=1: for n from 1 to 10 do Q[n]:=sort(expand(subs(x=1/t,Q[n-1])+(t*x+n-2)*subs(x=1,Q[n-1]))) od: for n from 0 to 10 do P[n]:=subs(x=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 are P(n,t)=Q(n,t,1), where Q(0,t,x)=1 and Q(n,t,x)=Q(n-1,t,1/t)+(tx+n-2)Q(n-1,t,1) for n>=1.

A121639 Number of 2-cell columns 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.

Original entry on oeis.org

0, 1, 5, 25, 147, 996, 7668, 66264, 635976, 6717600, 77482080, 969338880, 13076778240, 189261999360, 2925629280000, 48111515827200, 838731380659200, 15451544605593600, 299960798422118400, 6120505381423104000

Offset: 1

Emeric Deutsch, Aug 14 2006

nonn

a(n)=Sum(k*A121637(n,k), k=0..n-1).

			a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and only the vertical one has one 2-cell column.

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

Cf. A121637, A121555.

a[1]:=0: a[2]:=1: a[3]:=5: for n from 4 to 43 do a[n]:=n*a[n-1]+(n-1)!-(n-3)! od: seq(a[n],n=1..23);

a(1)=0, a(2)=1, a(3)=5, a(n)=na(n-1)+(n-1)!-(n-3)! for n>=4.

Conjecture D-finite with recurrence a(n) +(-2*n+1)*a(n-1) +n*(n-2)*a(n-2) +(2*n-7)*a(n-3) -(n-3)*(n-5)*a(n-4)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A121554 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (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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