A121637 -id:A121637 - cp's OEIS frontend

A121638 Number of deco polyominoes of height n, having no 2-cell columns. 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, 2, 7, 29, 147, 889, 6252, 50163, 452356, 4529812, 49878095, 598989496, 7791393260, 109129383735, 1637539745521, 26208427321596, 445652393850867, 8023380629061127, 152470440379483009, 3049854459983511047, 64054967040282793114, 1409361745326600931517

Offset: 1

Emeric Deutsch, Aug 14 2006

nonn

a(n) = A121637(n,0).

			a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and only the horizontal one has no 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, A001053.

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

D-finite with recurrence a(n)=(n-1)a(n-1)+a(n-3) for n>=3; a(1)=1, a(2)=1, a(3)=2.

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

A121638 Number of deco polyominoes of height n, having no 2-cell columns. 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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