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). 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.

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

0,4

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.

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

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

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

Sum(k*T(n,k),k>=0)=A181305(n).

References

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.

Links

Table of n, a(n) for n=0..58.

Formula

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

G.f. for column k is z^k*(1+z)(1-z)^2/(1-2z-2z^2+2z^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)].

Example

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;

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

Crossrefs

Cf. A003480, A181305, A181306, A181336

Sequence in context: A106479 A114422 A127501 * A339435 A118408 A079268

Adjacent sequences: A181301 A181302 A181303 * A181305 A181306 A181307

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 13 2010

Extensions

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

Status

approved