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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).
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%I #10 Aug 25 2024 13:35:24

%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 545,213,78,27,10,1,1,1023,4026,3279,1593,649,245,88,30,11,1,1,2047,9944,8981,4583,1932,748,277,98,33,12,1

%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

%t f[k_] := Sum[z^i, {i, 1, k}]/(1 - Sum[j*z^j, {j, 1, k}]);

%t T[n_, k_] := If[k <= n, SeriesCoefficient[f[k] - f[k-1], {z, 0, n}], 0];

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 25 2024, after Maple program *)

%Y Cf. A001519, A000225.

%K nonn,tabl

%O 1,5

%A _Emeric Deutsch_, Aug 04 2006