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

0,3

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.

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..66.

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.

Example

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;

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

Crossrefs

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

Sequence in context: A344913 A052079 A291483 * A166299 A182508 A213626

Adjacent sequences: A181292 A181293 A181294 * A181296 A181297 A181298

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 12 2010

Status

approved