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

0,3

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.

Links

Table of n, a(n) for n=0..24.

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).

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

Example

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.

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 *)

Prog

(PARI) a(n)=if(n, ([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 16, -20, 8]^(n-1)*[1; 5; 23; 99])[1, 1], 0) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A181330, A181294, A181292.

Sequence in context: A119012 A215038 A084615 * A268400 A364754 A339232

Adjacent sequences: A181328 A181329 A181330 * A181332 A181333 A181334

Keyword

nonn,easy,changed

Author

Emeric Deutsch, Oct 13 2010

Status

approved