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A181292 The sum of the entries in the top rows of all 2-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.

%I A181292 #12 Jul 24 2022 14:00:50
%S A181292 0,1,7,36,164,700,2868,11424,44576,171216,649520,2439360,9085632,
%T A181292 33605312,123561536,451998720,1646101504,5971400960,21586910976,
%U A181292 77796897792,279594972160,1002326793216,3585117623296,12796737085440
%N A181292 The sum of the entries in the top rows of all 2-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.
%H A181292 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 Journal of Combinatorics, 28, 2007, 1724-1741.
%H A181292 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,16,-4).
%F A181292 a(n) = Sum_{k=0..n} k*A059576(n,k).
%F A181292 G.f.: z(1-z)/(1-4z+2z^2)^2. [Corrected by _Georg Fischer_, May 19 2019]
%e A181292 a(2)=7 because the 2-compositions of 2, written as (top row / bottom row), are (0 / 2), (1 / 1), (2 / 0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), (0,0 / 1,1) and the sum of the entries in the top rows is 0 + 1 + 2 + 1 + 0 +0 +1 + 1 + 1 + 0 + 0 = 7.
%p A181292 g := z*(1-z)/(1-4*z+2*z^2)^2: gser := series(g, z = 0, 30): seq(coeff(gser, z, k), k = 0 .. 25);
%t A181292 CoefficientList[Series[x(1-x)/(1-4x+2x^2)^2, {x, 0, 30}], x] (* _Georg Fischer_, May 19 2019 *)
%Y A181292 Cf. A059576
%K A181292 nonn
%O A181292 0,3
%A A181292 _Emeric Deutsch_, Oct 12 2010