A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)).

1, 2, 6, 1, 20, 4, 64, 17, 1, 206, 68, 6, 662, 261, 32, 1, 2128, 976, 152, 8, 6840, 3577, 675, 51, 1, 21986, 12912, 2860, 280, 10, 70670, 46049, 11704, 1406, 74, 1, 227156, 162628, 46632, 6632, 460, 12, 730152, 569705, 181877, 29866, 2570, 101, 1, 2346942

Offset: 0

Author

Emeric Deutsch, Oct 12 2010

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
		

Links

• 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.

Crossrefs

Cf. A003480, A181300, A181301.

Programs

Maple

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

Formula

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).