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)).

1, 2, 6, 1, 18, 6, 54, 27, 1, 162, 108, 10, 486, 405, 64, 1, 1458, 1458, 334, 14, 4374, 5103, 1549, 117, 1, 13122, 17496, 6652, 760, 18, 39366, 59049, 27064, 4238, 186, 1, 118098, 196830, 105796, 21324, 1450, 22, 354294, 649539, 401041, 99646, 9480, 271

Offset: 0

Author

Emeric Deutsch, Oct 13 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.

Number of entries in row n is 1+floor(n/2).

Examples

			T(2,1) = 1 because we have (1/1) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
  1;
  2;
  6,1;
  18,6;
  54,27,1;
  162,108,10;
		

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, A008776, A054146.

Programs

Maple

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

Formula

G.f.: G(t,z) = (1-z)^2/(1-4*z+3*z^2-t*z^2).

G.f. of column k: z^(2*k)/((1-3*z)^(k+1)*(1-z)^(k-1)) (we have a Riordan array).

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

T(n,0) = 2*3^(n-1) = A008776(n-1).

Sum_{k>=0} k*T(n,k) = A054146(n-1).