A181329 Number of 2-compositions of n having no column with an even sum. 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.

1, 2, 4, 12, 32, 86, 232, 624, 1680, 4522, 12172, 32764, 88192, 237390, 638992, 1720000, 4629792, 12462194, 33544980, 90294348, 243048864, 654224230, 1761001208, 4740156528, 12759266608, 34344622042, 92446776092, 248842639740, 669819565056, 1802979787550, 4853151929120

Offset: 0

Author

Emeric Deutsch, Oct 13 2010

Keywords

• nonn
• easy
• changed

Comments

a(n) = A181327(n,0).

Number of compositions of n into odd parts where there is 2 sorts of part 1, 4 sorts of part 3, 6 sorts of part 5, ... , 2*k sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

Examples

			a(2)=4 because we have (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)).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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 (2,2,0,-1).

Crossrefs

Cf. A181327.

Programs

Maple

g := (1-z^2)^2/(1-2*z-2*z^2+z^4): gser := series(g, z = 0, 32): seq(coeff(gser, z, n), n = 0 .. 30);

Mathematica

CoefficientList[Series[(1 - x^2)^2/(1 - 2 x - 2 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2014 *)

PARI

Vec((1-z^2)^2/(1-2*z-2*z^2+z^4) + O(z^30)) \\ Stefano Spezia, Sep 05 2025

Formula

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