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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).

%I A121634 #9 Nov 15 2019 21:33:49
%S A121634 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 A121634 429,117,17,1,15840,12204,7366,3536,1131,219,23,1,146160,106308,63766,
%U A121634 32118,11453,2664,380,30,1,1491840,1036944,616436,320710,123742,32765,5704,620,38,1
%N 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).
%C A121634 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 A121634 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 A121634 Row sums are the factorials (A000142).
%F A121634 T(n,0) = A121635(n).
%F A121634 Sum_{k=0..n-1} k*T(n,k) = A121636(n).
%F A121634 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 A121634 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 A121634 Triangle starts:
%e A121634    1;
%e A121634    1,  1;
%e A121634    2,  3,  1;
%e A121634    8, 10,  5,  1;
%e A121634   42, 44, 25,  8,  1;
%p A121634 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 A121634 P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);
%t A121634 P[1, _] = 1; P[2, t_] = 1 + t;
%t A121634 Table[CoefficientList[P[n, t], t], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Nov 15 2019 *)
%Y A121634 Cf. A000142, A121635, A121636, A121585.
%K A121634 nonn,tabl
%O A121634 1,4
%A A121634 _Emeric Deutsch_, Aug 13 2006