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A121301 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).

%I A121301 #11 Aug 25 2024 16:15:29
%S A121301 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 A121301 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 A121301 10153,680,90,11,11,0,0,0,0,0,1,26875,1557,178,28,12,6,0,0,0,0,0,1,71021,3546,362,69,13,13,0,0,0,0,0,0,1
%N A121301 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 A121301 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 A121301 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 A121301 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 A121301 Triangle starts:
%e A121301   1;
%e A121301   1,1;
%e A121301   4,0,1;
%e A121301   10,2,0,1;
%e A121301   28,5,0,0,1;
%e A121301   75,10,3,0,0,1;
%e A121301   202,23,7,0,0,0,1;
%p A121301 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
%t A121301 f[k_] := z^k*(1 - z)/(z^2 - 2*z + 1 + z^(1 + k)*k - k*z^k - z^(1 + k));
%t A121301 T[n_, k_] := If[k <= n, SeriesCoefficient[f[k] - f[k+1], {z, 0, n}], 0];
%t A121301 Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 25 2024, after Maple Program *)
%Y A121301 Cf. A001519, A121469, A121300.
%K A121301 nonn,tabl
%O A121301 1,4
%A A121301 _Emeric Deutsch_, Aug 04 2006