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

Author

Emeric Deutsch, Oct 12 2010

Keywords

• nonn
• easy

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.

Examples

			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)).
		

Links

• 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).

Crossrefs

Cf. A181295, A181297, A181298, A181302.

Programs

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 *)

Formula

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

E.g.f.: exp(-x)*(exp(3*x)*(16*(1 + 7*x)*cosh(sqrt(2)*x) + sqrt(2)*(18 + 77*x)*sinh(sqrt(2)*x)) - 16)/98. - Stefano Spezia, May 11 2025

Extensions

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