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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.

%I A181329 #19 Sep 05 2025 14:55:42
%S A181329 1,2,4,12,32,86,232,624,1680,4522,12172,32764,88192,237390,638992,
%T A181329 1720000,4629792,12462194,33544980,90294348,243048864,654224230,
%U A181329 1761001208,4740156528,12759266608,34344622042,92446776092,248842639740,669819565056,1802979787550,4853151929120
%N 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.
%C A181329 a(n) = A181327(n,0).
%C A181329 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
%H A181329 Vincenzo Librandi, <a href="/A181329/b181329.txt">Table of n, a(n) for n = 0..1000</a>
%H A181329 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.
%H A181329 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,0,-1).
%F A181329 G.f.: (1-z^2)^2/(1-2*z-2*z^2+z^4).
%e A181329 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)).
%p A181329 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);
%t A181329 CoefficientList[Series[(1 - x^2)^2/(1 - 2 x - 2 x^2 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 05 2014 *)
%o A181329 (PARI) Vec((1-z^2)^2/(1-2*z-2*z^2+z^4) + O(z^30)) \\ _Stefano Spezia_, Sep 05 2025
%Y A181329 Cf. A181327.
%K A181329 nonn,easy,changed
%O A181329 0,2
%A A181329 _Emeric Deutsch_, Oct 13 2010