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A105929 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having k columns of height 1 starting at level 0.

%I A105929 #9 Jul 01 2025 09:27:43
%S A105929 1,0,1,1,0,1,2,2,0,1,6,3,3,0,1,16,9,4,4,0,1,43,22,13,5,5,0,1,114,58,
%T A105929 30,18,6,6,0,1,301,151,79,40,24,7,7,0,1,792,396,202,107,52,31,8,8,0,1,
%U A105929 2080,1038,526,270,143,66,39,9,9,0,1,5456,2722,1370,701,358,188,82,48,10,10,0,1
%N A105929 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having k columns of height 1 starting at level 0.
%C A105929 T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at height 1. Example: T(4,2)=3 because we have UDUDUUDD, UDUUDDUD and UUDDUDUD, where U=(1,1) and D=(1,-1). sum(T(n,k),k=0..n)=fibonacci(2n-1) (A001519). sum(k*T(n,k),k=0..n)=fibonacci(2n-1) (A001519). T(n,0)=A027994(n-2) for n>=2.
%H A105929 Elena Barcucci, Alberto del Lungo, S. Fezzi, and Renzo 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 A105929 Emeric Deutsch and Helmut 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 A105929 G.f.=(1-2z)^2/[(1-3z+z^2)(1-z-z^2-tz+tz^2)].
%e A105929 Triangle begins:
%e A105929   1;
%e A105929   0,1;
%e A105929   1,0,1;
%e A105929   2,2,0,1;
%e A105929   6,3,3,0,1;
%p A105929 G:=(1-2*z)^2/(1-3*z+z^2)/(1-z-z^2-t*z+t*z^2):Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..n+1) od;# yields sequence in triangular form
%Y A105929 Cf. A001519, A027994.
%K A105929 nonn,tabl
%O A105929 0,7
%A A105929 _Emeric Deutsch_, Apr 26 2005