A181296 The number of odd entries in all the 2-compositions of n.

0, 2, 10, 48, 208, 864, 3472, 13640, 52664, 200616, 755992, 2823688, 10468856, 38570504, 141341944, 515532424, 1872673144, 6777925768, 24453094264, 87966879368, 315629269368, 1129834372744, 4035747287416, 14387491636872

Offset

0,2

Comments

Also: Number of columns with distinct entries in all compositions of 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.

Links

Table of n, a(n) for n=0..23.

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.

Index entries for linear recurrences with constant coefficients, signature (7,-12,-4,12,-4).

Formula

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

a(n) = Sum_{k=0..n} k*A181295(n,k) = Sum_{k=0..n} k*A181302(n,k).

a(n) = 2*A181305(n). [R. J. Mathar, Oct 28 2010]

a(n) = 7*a(n-1)- 12*a(n-2)- 4*a(n-3)+12*a(n-4)-4*a(n-5). [Harvey P. Dale, Nov 11 2011]

Example

a(2)=10 because in the 2-compositions of 2, namely (1/1),(0/2),(2/0), (1,0/0,1),(0,1/1,0),(1,1/0,0), and (0,0/1,1), we have 2+0+0+2+2+2+2=10 odd entries (the 2-compositions are written as (top row / bottom row)).
a(1)=2 because in (0/1) and (1/0) we have a total of 2 columns with distinct entries (the 2-compositions are written as (top row / bottom row).

Maple

g := 2*z*(1-z)^2/((1+z)*(1-4*z+2*z^2)^2): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 25);

Mathematica

CoefficientList[Series[(2x (1-x)^2)/((1+x)(1-4x+2x^2)^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -12, -4, 12, -4}, {0, 2, 10, 48, 208}, 30] (* Harvey P. Dale_, Nov 11 2011 *)

Crossrefs

Cf. A181295, A181297, A181298, A181302.

Sequence in context: A116194 A054138 A192696 * A239073 A065982 A114693

Adjacent sequences: A181293 A181294 A181295 * A181297 A181298 A181299

Keyword

nonn

Author

Emeric Deutsch, Oct 12 2010

Extensions

Merged with a definition concerning row sums of A181302 - R. J. Mathar, Oct 28 2010

Status

approved