A181295 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k odd 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, 2, 0, 5, 0, 12, 0, 12, 7, 0, 46, 0, 29, 0, 58, 0, 152, 0, 70, 24, 0, 297, 0, 466, 0, 169, 0, 256, 0, 1236, 0, 1364, 0, 408, 82, 0, 1632, 0, 4575, 0, 3870, 0, 985, 0, 1072, 0, 8160, 0, 15702, 0, 10736, 0, 2378, 280, 0, 8160, 0, 35320, 0, 51121, 0, 29282, 0, 5741, 0

Offset: 0

Author

Emeric Deutsch, Oct 12 2010

Keywords

• nonn
• tabl

Comments

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

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

T(2n,0)=A003480(n).

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

T(n,n)=A000129(n+1) (the Pell numbers).

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

For the statistics "number of even entries" see A181297.

Examples

			T(2,2)=5 because we have (1/1),(1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1); the 2-compositions are written as (top row / bottom row).
Triangle starts:
1;
0,2;
2,0,5;
0,12,0,12;
7,0,46,0,29;
		

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.

Crossrefs

Cf. A003480, A000129, A181296, A181297, A181298,

Programs

Maple

G := (1-z^2)^2/(1-4*z^2+2*z^4-2*t*z-t^2*z^2): 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], t, k), k = 0 .. n) end do; # yields sequence in triangular form

Formula

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

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.