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A181306 Number of 2-compositions of n having no increasing columns.

%I A181306 #19 Jun 05 2025 20:37:53
%S A181306 1,1,3,7,18,44,110,272,676,1676,4160,10320,25608,63536,157648,391152,
%T A181306 970528,2408064,5974880,14824832,36783296,91266496,226449920,
%U A181306 561866240,1394099328,3459031296,8582528768,21294921472,52836837888,131098461184
%N A181306 Number of 2-compositions of n having no increasing columns.
%C A181306 Also, number of 2-compositions of n that have no odd entries in the top row.
%C A181306 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 A181306 From _Sean A. Irvine_, Jun 05 2025: (Start)
%C A181306 Also, for n>=1 the number of walks of length n-1 starting at vertex 1 in the graph:
%C A181306     1-2
%C A181306    /| |
%C A181306   0 | |
%C A181306    \| |
%C A181306     4-3. (End)
%H A181306 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="https://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 A181306 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A181306 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2).
%F A181306 G.f.: (1+z)*(1-z)^2/(1-2*z-2*z^2+2*z^3).
%F A181306 a(n) = A181304(n,0).
%F A181306 a(0)=1, a(1)=1, a(2)=3, a(3)=7, a(n)=2*a(n-1)+2*a(n-2)-2*a(n-3). - _Harvey P. Dale_, Mar 07 2012
%e A181306 a(2) = 3 because we have (1/1), (2/0), and (1,1/0,0) (the 2-compositions are written as (top row / bottom row)).
%e A181306 Alternatively, a(2) = 3 because we have (0/2), (2,0), and (0,0/1,1) (the 2-compositions are written as (top row/bottom row)).
%p A181306 g := (1+z)*(1-z)^2/(1-2*z-2*z^2+2*z^3): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
%t A181306 CoefficientList[Series[((1+x)(1-x)^2)/(1-2x-2x^2+2x^3),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{2,2,-2},{1,3,7},30]] (* _Harvey P. Dale_, Mar 07 2012 *)
%Y A181306 Cf. A181304.
%K A181306 nonn,easy
%O A181306 0,3
%A A181306 _Emeric Deutsch_, Oct 13 2010
%E A181306 Edited by _N. J. A. Sloane_, Oct 15 2010