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A121698 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an even level (1<=k<=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.

%I A121698 #11 May 05 2025 05:21:16
%S A121698 1,1,1,2,2,2,6,8,7,3,16,36,37,23,8,62,172,220,166,80,20,230,844,1383,
%T A121698 1338,835,338,72,1114,4796,9331,10828,8265,4282,1452,252,5268,27450,
%U A121698 64612,91023,85248,55445,25158,7524,1152,30702,181606,489847,798355
%N A121698 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an even level (1<=k<=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.
%C A121698 Row sums are the factorials (A000142).
%C A121698 T(n,0) = A121753(n).
%C A121698 Sum_{k=0..n-1} k*T(n,k) = A121754(n).
%H A121698 Elena Barcucci, Sara Brunetti, and Francesco Del Ristoro, <a href="http://www.numdam.org/item?id=ITA_2000__34_1_1_0">Succession rules and deco polyominoes</a>, Theoret. Informatics Appl., 34, 2000, 1-14.
%H A121698 Elena Barcucci, Alberto Del Lungo, and Renzo 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 A121698 The row generating polynomials P[n](s) are given by P[n](s) = Q[n](1,s), where Q[n](t,s) are defined by Q[n](t,s) = Q[n-1](s,t)+(floor(n/2)*t+floor((n-1)/2)*s)*Q[n-1](t,s) for n>=2 and Q[1](t,s) = t.
%e A121698 T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having 0 and 1 columns ending at an even level, respectively.
%e A121698 Triangle starts:
%e A121698   1;
%e A121698   1,1;
%e A121698   2,2,2;
%e A121698   6,8,7,3;
%e A121698   16,36,37,23,8;
%e A121698   62,172,220,166,80,20;
%p A121698 Q[1]:=t: for n from 2 to 10 do Q[n]:=expand(subs({t=s,s=t},Q[n-1])+(t*floor(n/2)+s*floor((n-1)/2))*Q[n-1]) od: for n from 1 to 10 do P[n]:=sort(subs(t=1,Q[n])) od: for n from 0 to 10 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form
%Y A121698 Cf. A000142, A121753, A121754, A121697.
%K A121698 nonn,tabl
%O A121698 1,4
%A A121698 _Emeric Deutsch_, Aug 23 2006