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A136129 Triangle read by rows: T(n,k) is the number of directed, vertically convex polyominoes of height n and area k (n<= k <=n(n+1)/2).

%I A136129 #5 Dec 11 2016 04:56:07
%S A136129 1,0,2,1,0,0,4,5,3,1,0,0,0,8,15,17,15,9,4,1,0,0,0,0,16,39,59,75,78,67,
%T A136129 48,29,14,5,1,0,0,0,0,0,32,95,175,269,358,419,432,400,334,250,166,97,
%U A136129 49,20,6,1,0,0,0,0,0,0,64,223,479,845,1300,1801,2269,2622,2805,2794,2593
%N A136129 Triangle read by rows: T(n,k) is the number of directed, vertically convex polyominoes of height n and area k (n<= k <=n(n+1)/2).
%C A136129 Row n contains n(n+1)/2 terms. Row sums yield A007808. Column sums yield the odd-indexed Fibonacci numbers (A001519).
%H A136129 E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, <a href="http://www.mat.univie.ac.at/~slc/opapers/s31barc.html">La hauteur des polyominos dirigés verticalement convexes</a>, Séminaire Lotharingien de Combinatoire, B31d (1993), 11 pp.
%F A136129 G.f. G(t,z) satisfies G(t,z)=zt(1-t)/(1-t-2zt+zt^2) +z(z-1)t^2*G(t,tz)/[(1-t-2zt+zt^2)(1-zt)]
%e A136129 Triangle starts:
%e A136129 1;
%e A136129 0,2,1;
%e A136129 0,0,4,5,3,1;
%e A136129 0,0,0,8,15,17,15,9,4,1;
%e A136129 0,0,0,0,16,39,59,75,78,67,48,29,14,5,1;
%p A136129 A:=t*z*(1-t)/(1-t-2*t*z+t^2*z): B:=t^2*z*(z-1)/((1-t-2*t*z+t^2*z)*(1-t*z)): Aser:=simplify(series(A,z=0,12)): Bser:=simplify(series(B,z=0,12)): for n to 12 do A[n]:=coeff(Aser,z,n): B[n]:=coeff(Bser,z,n) end do: P[1]:=A[1]: for n from 2 to 7 do P[n]:=sort(expand(simplify(A[n]+add(B[n-j]*P[j]*t^j,j=1..n-1)))) end do: for n to 7 do seq(coeff(P[n],t,j),j=1..(1/2)*n*(n+1)) end do;
%Y A136129 Cf. A007808, A001519.
%K A136129 nonn,tabf
%O A136129 1,3
%A A136129 _Emeric Deutsch_, Jan 21 2008