A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even 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, 0, 6, 0, 8, 0, 16, 3, 0, 35, 0, 44, 0, 28, 0, 132, 0, 120, 8, 0, 160, 0, 460, 0, 328, 0, 92, 0, 748, 0, 1528, 0, 896, 21, 0, 642, 0, 3117, 0, 4916, 0, 2448, 0, 290, 0, 3552, 0, 12062, 0, 15456, 0, 6688, 55, 0, 2380, 0, 17119, 0, 44318, 0, 47760, 0, 18272, 0, 888, 0

Offset

0,3

Comments

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

T(2n-1,0)=0.

T(2n,0)=A000045(2n) (Fibonacci numbers).

T(n,k)=0 if n and k have opposite parities.

T(n,n)=A002605(n+1).

Sum(k*T(n,k),k=0..n)=A181298.

For the statistics "number of odd entries" see A181295.

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..68.

Formula

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

The g.f. H(t,s,z), where z marks the size of the 2-composition and t (s) marks the number of odd (even) entries, is H=1/(1-h), where h=z(t+sz)(2s+tz-sz^2)/(1-z^2)^2.

Example

T(2,2)=6 because we have (0 / 2), (2 / 0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), (0,0 / 1,1) (the 2-compositions are written as (top row / bottom row).
Triangle starts:
1;
0,2;
1,0,6;
0,8,0,16;
3,0,35,0,44;

Maple

G := (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], s, k), k = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A003480, A000045, A181295, A181296, A181298.

Sequence in context: A266904 A299198 A137477 * A196776 A336345 A157982

Adjacent sequences: A181294 A181295 A181296 * A181298 A181299 A181300

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 12 2010

Status

approved