%I #12 Jul 22 2017 08:56:12

%S 1,2,4,1,8,5,16,17,1,32,49,8,64,129,39,1,128,321,150,11,256,769,501,

%T 70,1,512,1793,1524,338,14,1024,4097,4339,1375,110,1,2048,9217,11762,

%U 4973,640,17,4096,20481,30705,16508,3075,159,1,8192,45057,77808,51340,12918

%N Triangle read by rows: T(n,k) = is the number of directed column-convex polyominoes of area 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 Also number of nondecreasing Dyck paths of semilength n and such that there are k positive differences in the sequence of the valley altitudes, preceded by a 0. Example: T(5,2)=1 because we have UUDUUDUDDD, where U=(1,1) and D=(1,-1) (the valleys are at the altitudes 1 and 2 with two "jumps" in the sequence 0,1,2).

%C Row n has ceiling(n/2) terms.

%C Row sums are the odd-subscripted Fibonacci numbers (A001519).

%H E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.

%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.

%H E. Deutsch and H. Prodinger, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00222-6">A bijection between directed column-convex polyominoes and ordered trees of height at most three</a>, Theoretical Comp. Science, 307, 2003, 319-325.

%F T(n,0) = 2^(n-1) = A000079(n-1).

%F T(n,1) = 1 + (n-3)*2^(n-2) = A000337(n-2).

%F T(n,2) = A055581(n-5).

%F Sum_{k=0..ceiling(n/2)-1} k*T(n,k) = A001870(n-3).

%F T(n,k) = Sum_{j=0..n-2*k-1} 2^j*binomial(n-k-2-j,k-1)*binomial(k+j,k) for k >= 1; T(n,0) = 2^(n-1).

%F G.f.: G(t,z) = z(1-z)/(1-3z+2z^2-tz^2).

%e T(5,2)=1 because we have the directed column-convex polyomino [(0,2),(1,3),(2,3)] (here the j-th pair gives the lower and upper levels of the j-th column).

%e Triangle starts:

%e 1;

%e 2;

%e 4, 1;

%e 8, 5;

%e 16, 17, 1;

%e 32, 49, 8;

%e 64, 129, 39, 1;

%p with(combinat): T:=(n,k)->add(2^j*binomial(n-k-2-j,k-1)*binomial(k+j,k),j=0..n-2*k-1): for n from 0 to 15 do seq(T(n,k),k=0..ceil(n/2)-1) od; # yields sequence in triangular form

%Y Cf. A000079, A000337, A001519, A001870, A055581.

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Aug 02 2006