A121581 -id:A121581 - cp's OEIS frontend

A121582 Number of cells in column 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.

0, 1, 7, 40, 252, 1837, 15259, 141798, 1455694, 16360387, 199845957, 2637020884, 37388864368, 566971338009, 9157693715407, 156975522127762, 2846305448882274, 54432896145210943, 1095019542858729769

Offset: 1

Emeric Deutsch, Aug 11 2006

nonn

a(n)=Sum(k*A121581(n,k),k=0..n-1).

			a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.

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

Harvey P. Dale, Table of n, a(n) for n = 1..400

Cf. A121580, A121581, A121584.

a:=proc(n) if n=1 then 0 elif n=2 then 1 else ((2*n-3)*a(n-1)-(n-1)*a(n-2)+(n-1)!*(n-2)*(n^2-3*n+4)/2)/(n-2) fi end: seq(a(n),n=1..22);

RecurrenceTable[{a[1]==0,a[2]==1,a[n]==((2n-3)a[n-1]-(n-1)a[n-2]+ (n-1)!(n-2) (n^2-3n+4)/2)/(n-2)},a,{n,20}] (* Harvey P. Dale, Oct 23 2012 *)

a(1)=0, a(2)=1, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)+(n-1)!(n-2)(n^2-3n+4)/2]/(n-2) for n>=3.
a(n) ~ n*n!/2. - Vaclav Kotesovec, Aug 15 2013
D-finite with recurrence (-49*n+454)*a(n) +(49*n^2-454*n-1328)*a(n-1) +(49*n^2+1553*n-1464)*a(n-2) +(-581*n^2+612*n+1035)*a(n-3) +(819*n^2-3682*n+4007)*a(n-4) -4*(84*n-169)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A121583 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the first two columns (n>=1, k>=1). 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, 0, 2, 0, 1, 4, 1, 0, 2, 6, 10, 5, 1, 0, 6, 16, 29, 34, 23, 11, 1, 0, 24, 60, 102, 148, 154, 119, 77, 35, 1, 0, 120, 288, 474, 668, 867, 874, 719, 533, 341, 155, 1, 0, 720, 1680, 2712, 3768, 4834, 5906, 5914, 5039, 4013, 2957, 1901, 875, 1, 0, 5040, 11520, 18360

Offset: 1

Emeric Deutsch, Aug 11 2006

Row n has 2n-2 terms (n>=2). Row sums are the factorials (A000142). Sum(k*T(n,k), k=0..n)=A121584(n)

			T(2,2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having 2 cells in their first two columns.
Triangle starts:
1;
0,2;
0,1,4,1;
0,2,6,10,5,1;
0,6,16,29,34,23,11,1;

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

Cf. A000142, A100822, A121581, A121584.

Q[1]:=t: for n from 2 to 9 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))) od: for n from 1 to 9 do P[n]:=sort(subs(s=t,Q[n])): od: 1; for n from 1 to 9 do seq(coeff(P[n],t,j),j=1..2*n-2) od; # yields sequence in triangular form

The generating polynomial of row n is P(n,t)=Q(n,t,t), 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.

cp's OEIS Frontend

A121582 Number of cells in column 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.

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