0,2

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.

Row n contains 1+floor(n/2) entries.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

G.f.: G(t,z) = (1+z)*(1-z)^2/(1-3*z-z^2+z^3-t*(1-z)*z^2).

Sum_{k>=0} T(n,k) = A003480(n).

T(n,0) = A181301(n).

Sum_{k>=0} k*T(n,k) = A181300(n).

T(3,1) = 4 because we have (1,1/1,0), (1,0/1,1), (1,1/0,1), (0,1/1,1) (the 2-compositions are written as (top row/bottom row)).

Triangle starts:

1;

2;

6,1;

20,4;

64,17,1;

G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Cf. A003480, A181300, A181301.

Sequence in context: A181307 A008855 A345043 * A181365 A221913 A280370

Adjacent sequences: A181296 A181297 A181298 * A181300 A181301 A181302

nonn,tabf

Emeric Deutsch, Oct 12 2010

approved