A121635 -id:A121635 - cp's OEIS frontend

A121634 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n >= 1; 0 <= k <= n-1).

1, 1, 1, 2, 3, 1, 8, 10, 5, 1, 42, 44, 25, 8, 1, 264, 242, 144, 57, 12, 1, 1920, 1594, 962, 429, 117, 17, 1, 15840, 12204, 7366, 3536, 1131, 219, 23, 1, 146160, 106308, 63766, 32118, 11453, 2664, 380, 30, 1, 1491840, 1036944, 616436, 320710, 123742, 32765, 5704, 620, 38, 1

Offset: 1

Emeric Deutsch, Aug 13 2006

A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

			T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 columns with exactly 2 cells starting at level 0.
Triangle starts:
   1;
   1,  1;
   2,  3,  1;
   8, 10,  5,  1;
  42, 44, 25,  8,  1;

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

Cf. A000142, A121635, A121636, A121585.

P[1]:=1: P[2]:=1+t: for n from 3 to 11 do P[n]:=sort(expand((t+n-2)*((n-2)!+P[n-1]))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);
P[1, ] = 1; P[2, t] = 1 + t;
Table[CoefficientList[P[n, t], t], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Row sums are the factorials (A000142).
T(n,0) = A121635(n).
Sum_{k=0..n-1} k*T(n,k) = A121636(n).
The row generating polynomials satisfy P(n,t) = (t+n-2)[(n-2)!+P(n-1,t)] for n >= 3, P(1,t)=1 and P(2,t)=1+t.

A269722 Triangle read by rows: coefficients of polynomials arising as traces of certain web-coloring matrices.

Original entry on oeis.org

1, 2, 2, 6, 8, 6, 24, 30, 42, 24, 120, 116, 216, 264, 120, 720, 532, 1002, 1920, 1920, 720, 5040, 2848, 4626, 11688, 19200, 15840, 5040

Offset: 1

N. J. A. Sloane, Mar 04 2016

			Triangle begins:
     1
     2    2
     6    8    6
    24   30   42    24
   120  116  216   264   120
   720  532 1002  1920  1920   720
  5040 2848 4626 11688 19200 15840 5040
  ...

Mark Dukes, Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016. See Fig. 7 p. 19.
Mark Dukes, Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.

Outer diagonals are A000142. Another diagonal is A121635.

Cf. A281351.

cp's OEIS Frontend

A121634 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n >= 1; 0 <= k <= n-1).

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