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A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (0<=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 A121697 #9 May 05 2025 05:33:42
%S A121697 1,0,1,1,0,1,2,2,1,1,4,8,7,3,2,14,32,37,23,10,4,44,142,207,180,97,38,
%T A121697 12,194,730,1267,1327,911,425,150,36,812,3810,8104,10387,8876,5257,
%U A121697 2222,708,144,4362,23284,56987,84792,85317,60814,31368,11972,3408,576,22716
%N A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (0<=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 A121697 Row sums are the factorials (A000142).
%C A121697 T(n,0) = A121751(n), T(n,n) = A010551(n-1) for n>=1.
%C A121697 Sum_{k=0..n} k*T(n,k) = A121752(n).
%H A121697 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 A121697 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 A121697 The row generating polynomials P[n](t) are given by P[n](t) = Q[n](t,1), 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[0](t,s) = 1, Q[1](t,s) = t.
%e A121697 T(2,0)=1, T(2,1)=0 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns ending at an odd level, respectively.
%e A121697 Triangle starts:
%e A121697   1;
%e A121697   0,1;
%e A121697   1,0,1;
%e A121697   2,2,1,1;
%e A121697   4,8,7,3,2;
%e A121697   14,32,37,23,10,4;
%p A121697 Q[0]:=1: 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 0 to 10 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A121697 Cf. A000142, A010551, A121751, A121752, A121698.
%K A121697 nonn,tabl
%O A121697 0,7
%A A121697 _Emeric Deutsch_, Aug 23 2006