A181336 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even 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.

1, 1, 1, 2, 4, 1, 5, 11, 7, 1, 11, 31, 29, 10, 1, 25, 83, 102, 56, 13, 1, 56, 217, 329, 245, 92, 16, 1, 126, 556, 1000, 938, 487, 137, 19, 1, 283, 1403, 2917, 3292, 2180, 855, 191, 22, 1, 636, 3498, 8247, 10865, 8740, 4406, 1376, 254, 25, 1, 1429, 8636, 22756, 34248

Offset

0,4

Comments

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

T(n,0)=A006054(n+1) (n>=1).

Sum(k*T(n,k), k>=0)=A181337(n).

For the statistic "number of odd entries in the top row" see A181304.

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(s,z)=(1+z)(1-z)^2/[1-2z-z^2+z^3-sz(1+z-z^2)].

The g.f. of column k is z^k*(1+z)(1-z)^2*(1+z-z^2)^k/(1-2z-z^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)].

Example

T(2,1)=4 because we have (0/2), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
5,11,7,1;
11,31,29,10,1;
25,83,102,56,13,1;

Maple

G := (1+z)*(1-z)^2/(1-2*z-z^2+z^3-s*z*(1+z-z^2)): Gser := simplify(series(G, z = 0, 13)): 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], s, k), k = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A003480, A006054, A181304, A181337,

Sequence in context: A118906 A085059 A336164 * A238731 A124037 A126126

Adjacent sequences: A181333 A181334 A181335 * A181337 A181338 A181339

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 14 2010

Status

approved