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%I #5 Feb 14 2016 05:42:54
%S 1,1,1,1,3,1,1,7,4,1,1,15,12,5,1,1,31,35,15,6,1,1,63,95,48,18,7,1,1,
%T 127,249,145,58,21,8,1,1,255,640,418,181,68,24,9,1,1,511,1615,1180,
%U 545,213,78,27,10,1,1,1023,4026,3279,1593,649,245,88,30,11,1,1,2047,9944,8981
%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 longest column (1<=k<=n).
%C Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=1; T(n,2)=2^(n-1)-1=A000225(n-1); 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]=Sum(z^i, i=1..k)/[1-Sum(jz^j, j=1..k)] is the g.f. for directed column-convex polyominoes whose columns have height at most k.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,3,1;
%e 1,7,4,1;
%e 1,15,12,5,1;
%e 1,31,35,15,6,1;
%p f:=k->sum(z^i,i=1..k)/(1-sum(j*z^j,j=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 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%Y Cf. A001519, A000225.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Aug 04 2006
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