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)). 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, 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

0,2

Comments

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

Sum of entries in row n is A003480(n).

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

Sum(k*T(n,k),k>=0)=A054146(n-1).

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..47.

Formula

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

The g.f. of column k is z^{2k}/[(1-3z)^{k+1}*(1-z)^{k-1}] (we have a Riordan array).

Example

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;

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

Crossrefs

Cf. A003480, A008776, A054146.

Sequence in context: A281307 A281417 A175353 * A008855 A345043 A181299

Adjacent sequences: A181304 A181305 A181306 * A181308 A181309 A181310

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 13 2010

Status

approved