A181302 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns with distinct 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, 0, 2, 1, 2, 4, 0, 8, 8, 8, 2, 8, 32, 24, 16, 0, 24, 56, 104, 64, 32, 4, 24, 152, 248, 304, 160, 64, 0, 64, 248, 712, 896, 832, 384, 128, 8, 64, 568, 1496, 2800, 2880, 2176, 896, 256, 0, 160, 888, 3560, 6976, 9824, 8576, 5504, 2048, 512, 16, 160, 1848, 6904, 17904

Offset

0,3

Comments

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

Sum(k*T(n,k),k>=0)=A181296(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..59.

Formula

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

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

Example

T(2,1)=2 because we have (0/2) and (2/0) (the 2-compositions are written as (top row/bottom row).
Triangle starts:
1;
0,2;
1,2,4;
0,8,8,8;
2,8,32,24,16;

Maple

G := (1+z)*(1-z)^2/((1-z)*(1-2*z^2)-2*t*z): 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, A181296.

Sequence in context: A131022 A137408 A007461 * A304785 A143446 A110330

Adjacent sequences: A181299 A181300 A181301 * A181303 A181304 A181305

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 13 2010

Status

approved