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A181307 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)).

%I A181307 #8 May 11 2025 09:14:00
%S A181307 1,2,6,1,18,6,54,27,1,162,108,10,486,405,64,1,1458,1458,334,14,4374,
%T A181307 5103,1549,117,1,13122,17496,6652,760,18,39366,59049,27064,4238,186,1,
%U A181307 118098,196830,105796,21324,1450,22,354294,649539,401041,99646,9480,271
%N A181307 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)).
%C A181307 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 A181307 Number of entries in row n is 1+floor(n/2).
%H A181307 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 A181307 G.f.: G(t,z) = (1-z)^2/(1-4*z+3*z^2-t*z^2).
%F A181307 G.f. of column k: z^(2*k)/((1-3*z)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).
%F A181307 Sum_{k>=0} T(n,k) = A003480(n).
%F A181307 T(n,0) = 2*3^(n-1) = A008776(n-1).
%F A181307 Sum_{k>=0} k*T(n,k) = A054146(n-1).
%e A181307 T(2,1) = 1 because we have (1/1) (the 2-compositions are written as (top row / bottom row)).
%e A181307 Triangle starts:
%e A181307   1;
%e A181307   2;
%e A181307   6,1;
%e A181307   18,6;
%e A181307   54,27,1;
%e A181307   162,108,10;
%p A181307 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 A181307 Cf. A003480, A008776, A054146.
%K A181307 nonn,tabf
%O A181307 0,2
%A A181307 _Emeric Deutsch_, Oct 13 2010