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%I #2 Mar 30 2012 17:36:24
%S 1,2,6,1,18,6,54,27,1,162,108,10,486,405,64,1,1458,1458,334,14,4374,
%T 5103,1549,117,1,13122,17496,6652,760,18,39366,59049,27064,4238,186,1,
%U 118098,196830,105796,21324,1450,22,354294,649539,401041,99646,9480,271
%N Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with only nonzero entries (0<=k<=floor(n/2)). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
%C Number of entries in row n is 1+floor(n/2).
%C Sum of entries in row n is A003480(n).
%C T(n,0)=2*3^{n-1}=A008776(n-1).
%C Sum(k*T(n,k),k>=0)=A054146(n-1).
%D G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
%F G.f. = G(t,z)=(1-z)^2/(1-4z+3z^2-tz^2).
%F The g.f. of column k is z^{2k}/[(1-3z)^{k+1}*(1-z)^{k-1}] (we have a Riordan array).
%e T(2,1)=1 because we have (1/1) (the 2-compositions are written as (top row / bottom row).
%e Triangle starts:
%e 1;
%e 2;
%e 6,1;
%e 18,6;
%e 54,27,1;
%e 162,108,10;
%p G := (1-z)^2/(1-4*z+3*z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y Cf. A003480, A008776, A054146.
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, Oct 13 2010
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