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A121579 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n>=1, k>=0).

%I A121579 #8 Jul 22 2017 08:35:26
%S A121579 1,2,5,1,16,8,65,52,3,326,344,50,1957,2473,595,15,13700,19676,6524,
%T A121579 420,109601,173472,71862,7840,105,986410,1686912,823836,127232,4410,
%U A121579 9864101,17981193,9976686,1975750,118125,945,108505112,208769296,128350992
%N A121579 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n>=1, k>=0).
%C A121579 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 A121579 Row n contains ceiling(n/2) terms.
%C A121579 Row sums are the factorials (A000142).
%C A121579 T(n,0) = A000522(n).
%C A121579 T(2n+1,n) = (2n-1)!! = A001147(n) (the double factorials).
%C A121579 Sum_{k=0..n} k*T(n,k) = A002538(n-2) for n >= 3.
%H A121579 E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%F A121579 The row generating polynomials are P(n,t) = Q(n,t,1), where Q(1,t,x) = 1 and Q(n,t,x) = Q(n-1,t,t) + (n-1)xQ(n-1,t,1) for n >= 2.
%e A121579 T(2,0)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along the lower contour.
%e A121579 Triangle starts:
%e A121579     1;
%e A121579     2;
%e A121579     5,   1;
%e A121579    16,   8;
%e A121579    65,  52,   3;
%e A121579   326, 344,  50;
%p A121579 Q[1]:=1: for n from 2 to 13 do Q[n]:=sort(expand(subs(x=t,Q[n-1])+(n-1)*x*subs(x=1,Q[n-1]))) od: for n from 1 to 13 do P[n]:=subs(x=1,Q[n]) od: for n from 1 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)-1) od; # yields sequence in triangular form
%Y A121579 Cf. A000142, A000522, A001147, A002538.
%K A121579 nonn,tabf
%O A121579 1,2
%A A121579 _Emeric Deutsch_, Aug 08 2006