A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=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

Author

Emeric Deutsch, Oct 12 2010

Keywords

• nonn
• tabl

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.

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

Examples

			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;
		

Links

• 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.

Crossrefs

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

Programs

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

Formula

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

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

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

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

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

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

Sum_{k=0..n} k*T(n,k) = A181298(n).