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%I #5 Feb 14 2016 05:42:58
%S 1,1,1,4,0,1,10,2,0,1,28,5,0,0,1,75,10,3,0,0,1,202,23,7,0,0,0,1,540,
%T 57,8,4,0,0,0,1,1440,129,18,9,0,0,0,0,1,3828,294,43,10,5,0,0,0,0,1,
%U 10153,680,90,11,11,0,0,0,0,0,1,26875,1557,178,28,12,6,0,0,0,0,0,1,71021,3546
%N Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and having k cells in the shortest column (1<=k<=n).
%C Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=fibonacci(2n-1)-A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n and having k 1-cell columns). T(n,n)=1.
%H E. Barcucci, R. Pinzani and R. Sprugnoli, <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%F G.f. of column k is f[k]-f[k+1], where f[k]=z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(1+k)) is the g.f. for directed column-convex polyominoes whose columns have height at least k.
%e Triangle starts:
%e 1;
%e 1,1;
%e 4,0,1;
%e 10,2,0,1;
%e 28,5,0,0,1;
%e 75,10,3,0,0,1;
%e 202,23,7,0,0,0,1;
%p f:=k->z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(1+k)): T:=proc(n,k) if k<=n then coeff(series(f(k)-f(k+1),z=0,15),z,n) else 0 fi end: for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%Y Cf. A001519, A121469, A121300.
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Aug 04 2006
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