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A181293 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's (0<=k<=n) 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.

%I A181293 #8 Sep 14 2015 03:19:07
%S A181293 1,0,2,1,2,4,2,6,8,8,4,14,24,24,16,8,32,64,80,64,32,16,72,164,240,240,
%T A181293 160,64,32,160,408,680,800,672,384,128,64,352,992,1848,2480,2464,1792,
%U A181293 896,256,128,768,2368,4864,7296,8288,7168,4608,2048,512,256,1664,5568
%N A181293 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's (0<=k<=n) 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 A181293 The sum of entries in row n is A003480(n).
%C A181293 Sum_{k=0..n} k*T(n,k) = A181294(n).
%H A181293 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.
%F A181293 G.f.: G(t,z) = (1-z)^2/(1-2*z-2*t*z+2*t*z^2).
%F A181293 G.f. of column k is 2^k*z^k*(1-z)^{k+2}/(1-2*z)^{k+1} (we have a Riordan array).
%e A181293 T(2,0)=1, T(2,1)=2, T(2,2)=4 because the 2-compositions of 2, written as (top row/bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1).
%e A181293 Triangle starts:
%e A181293 1;
%e A181293 0,2;
%e A181293 1,2,4;
%e A181293 2,6,8,8;
%e A181293 4,14,24,24,16;
%e A181293 ...
%p A181293 G := (1-z)^2/(1-2*z-2*t*z+2*t*z^2): Gser := simplify(series(G, z = 0, 14)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A181293 Cf. A003480, A181294.
%K A181293 nonn,tabl
%O A181293 0,3
%A A181293 _Emeric Deutsch_, Oct 12 2010