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).

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

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.

Links

Table of n, a(n) for n=0..59.

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.

Formula

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

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

Sum_{k>=0} k*T(n,k) = A181296(n).

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

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