A181331 Number of 0's in the top rows of all 2-compositions of n.

0, 1, 5, 23, 99, 408, 1632, 6388, 24596, 93488, 351664, 1311536, 4856432, 17873408, 65436544, 238480960, 865665600, 3131196672, 11290210560, 40594476800, 145588087552, 520933746688, 1860059009024, 6628828632064, 23582036472832

Offset: 0

Author

Emeric Deutsch, Oct 13 2010

Keywords

• nonn

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.

Examples

			a(2)=5 because in (0/2), (1/1), (2,0), (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)) we have 1+0+1+1+1+0+2=5 zeros.
		

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.
• Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-4).

Crossrefs

Cf. A181330, A181294.

Programs

Maple

g := z*(1-z)^3/(1-4*z+2*z^2)^2: gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

Mathematica

LinearRecurrence[{8, -20, 16, -4}, {0, 1, 5, 23, 99}, 25] (* Georg Fischer, Feb 01 2021 *)

Formula

a(n) = Sum_{k=0..n} A181330(n,k).

a(n) = (1/2)*A181294(n).

G.f.: x*(1 - x)^3 / (1 - 4*x + 2*x^2)^2.

a(n) = A181292(n)-2*A181292(n-1)+A181292(n-2). - R. J. Mathar, Jul 24 2022