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%I #3 Mar 30 2012 17:36:24
%S 1,0,2,1,2,4,0,8,8,8,2,8,32,24,16,0,24,56,104,64,32,4,24,152,248,304,
%T 160,64,0,64,248,712,896,832,384,128,8,64,568,1496,2800,2880,2176,896,
%U 256,0,160,888,3560,6976,9824,8576,5504,2048,512,16,160,1848,6904,17904
%N Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (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 The sum of entries in row n is A003480(n).
%C Sum(k*T(n,k),k>=0)=A181296(n).
%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)(1-z)^2/[(1-z)(1-2z^2)-2tz].
%F G.f. of column k is 2^k*z^k*(1+z)/[(1-2z^2)^{k+1}*(1-z)^{k-1}] (we have a Riordan array).
%e T(2,1)=2 because we have (0/2) and (2/0) (the 2-compositions are written as (top row/bottom row).
%e Triangle starts:
%e 1;
%e 0,2;
%e 1,2,4;
%e 0,8,8,8;
%e 2,8,32,24,16;
%p G := (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z): Gser := simplify(series(G, z = 0, 15)): 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 Cf. A003480, A181296.
%K nonn,tabl
%O 0,3
%A _Emeric Deutsch_, Oct 13 2010
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