A181304 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with increasing entries (0<=k<=n).

1, 1, 1, 3, 3, 1, 7, 11, 5, 1, 18, 33, 23, 7, 1, 44, 100, 87, 39, 9, 1, 110, 288, 310, 177, 59, 11, 1, 272, 820, 1036, 728, 311, 83, 13, 1, 676, 2288, 3338, 2768, 1450, 497, 111, 15, 1, 1676, 6316, 10416, 9976, 6172, 2588, 743, 143, 17, 1, 4160, 17244, 31752, 34448

Offset: 0

Author

Emeric Deutsch, Oct 13 2010

Keywords

• nonn
• tabl

Comments

Also, triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd entries in the top row. 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.

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.

The sum of entries in row n is A003480(n).

For the statistic "number of even entries in the top row" see A181336.

Examples

			T(2,1) = 3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Alternatively, T(2,1) = 3 because we have (1/1), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
  1;
  1,1;
  3,3,1;
  7,11,5,1;
  18,33,23,7,1;
  44,100,87,39,9,1;
		

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, A181305, A181306, A181336.

Programs

Maple

G := (1+z)*(1-z)^2/(1-(2+t)*z-2*z^2+2*z^3): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

Formula

T(n,0) = A181306(n).

Sum_{k>=0} k*T(n,k) = A181305(n).

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

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

The g.f. H=H(t,s,z), where z marks size and t (s) marks odd (even) entries in the top row, is given by H = (1+z)(1-z)^2/[(1+z)(1-z)^2-(t+s)z-sz^2*(1-z)].

Extensions

Edited by N. J. A. Sloane, Oct 15 2010