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A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (0<=k<=n).

%I A181302 #14 May 11 2025 09:14:19
%S A181302 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 A181302 160,64,0,64,248,712,896,832,384,128,8,64,568,1496,2800,2880,2176,896,
%U A181302 256,0,160,888,3560,6976,9824,8576,5504,2048,512,16,160,1848,6904,17904
%N A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct entries (0<=k<=n).
%C A181302 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.
%H A181302 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="https://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 A181302 G.f.: G(t,z) = (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z).
%F A181302 G.f. of column k: 2^k*z^k*(1+z)/((1-2*z^2)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
%F A181302 Sum_{k>=0} k*T(n,k) = A181296(n).
%F A181302 Sum_{k>=0} T(n,k) = A003480(n).
%e A181302 T(2,1) = 2 because we have (0/2) and (2/0) (the 2-compositions are written as (top row/bottom row)).
%e A181302 Triangle starts:
%e A181302   1;
%e A181302   0,2;
%e A181302   1,2,4;
%e A181302   0,8,8,8;
%e A181302   2,8,32,24,16;
%p A181302 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 A181302 Cf. A003480, A181296.
%K A181302 nonn,tabl
%O A181302 0,3
%A A181302 _Emeric Deutsch_, Oct 13 2010