%N Number of increasing columns in all 2-compositions of 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 Also, number of odd entries in the top rows of all 2-compositions of 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 a(n)=Sum(A181304(n,k), k=0..n).
%C For the case of the even entries see A181337.
%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.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7, -12, -4, 12, -4).
%F G.f. = z(1-z)^2/[(1+z)(1-4z+2z^2)^2].
%e a(1)=1 because in the 2-compositions of 1, namely (0/1) and (1/0) we have only one increasing column (the 2-compositions are written as (top row / bottom row).
%e a(2)=5 because in (0/2),(1/1),(2,0),(1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1) (the 2-compositions are written as (top row/bottom row)) we have 0+1+0+1+1+2+0=5 odd entries.
%p g := z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, k), k = 0 .. 27);
%Y Cf. A181304, A181337.
%A _Emeric Deutsch_, Oct 13 2010
%E Edited by _N. J. A. Sloane_, Oct 15 2010