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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).
2

%I #9 Nov 15 2019 21:33:49

%S 1,1,1,2,3,1,8,10,5,1,42,44,25,8,1,264,242,144,57,12,1,1920,1594,962,

%T 429,117,17,1,15840,12204,7366,3536,1131,219,23,1,146160,106308,63766,

%U 32118,11453,2664,380,30,1,1491840,1036944,616436,320710,123742,32765,5704,620,38,1

%N 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).

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

%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="https://doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.

%F Row sums are the factorials (A000142).

%F T(n,0) = A121635(n).

%F Sum_{k=0..n-1} k*T(n,k) = A121636(n).

%F 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.

%e 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.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 2, 3, 1;

%e 8, 10, 5, 1;

%e 42, 44, 25, 8, 1;

%p 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

%t P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);

%t P[1, _] = 1; P[2, t_] = 1 + t;

%t Table[CoefficientList[P[n, t], t], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Nov 15 2019 *)

%Y Cf. A000142, A121635, A121636, A121585.

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_, Aug 13 2006