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A181366 Least entry in a 2-composition of n, summed over 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 A181366 #13 Nov 15 2020 09:29:25
%S A181366 0,1,2,5,10,20,38,74,144,282,552,1086,2144,4247,8430,16761,33364,
%T A181366 66479,132566,264520,528078,1054636,2106854,4209853,8413548,16817253,
%U A181366 33618758,67212301,134384182,268703498,537302782,1074437977,2148606246
%N A181366 Least entry in a 2-composition of n, summed over 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.
%C A181366 a(n) = Sum(k*A181365(n,k), k>=0).
%H A181366 Alois P. Heinz, <a href="/A181366/b181366.txt">Table of n, a(n) for n = 1..500</a>
%H A181366 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.
%F A181366 a(n) = Sum_{k>=0} k*A181365(n,k).
%F A181366 G.f. for 2-compositions with all entries >= k is h(k,z)=(1-z)^2/(1-2z+z^2-z^{2k}) if k>0 and h(0,z)=(1-z)^2/(1-4z+2z^2) if k=0.
%F A181366 G.f. for 2-compositions with least entry k is f(k,z)=h(k,z)-h(k+1,z).
%F A181366 G.f.: G(z) = Sum(k*f(k,z), k=1..infinity).
%F A181366 a(n) ~ 2^(n-2). - _Vaclav Kotesovec_, Sep 03 2014
%e A181366 a(2)=1 because the 2-compositions of 2, written as (top row / bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1) and the least entries are 1 and eight 0's.
%p A181366 h := proc (k) if k = 0 then (1-z)^2/(1-4*z+2*z^2) else (1-z)^2/(1-2*z+z^2-z^(2*k)) end if end proc: f := proc (k) options operator, arrow; h(k)-h(k+1) end proc: G := sum(k*f(k), k = 1 .. 50): Gser := series(G, z = 0, 45): seq(coeff(Gser, z, n), n = 1 .. 35);
%t A181366 terms = 100;
%t A181366 A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[i==0 && j==0, 0, A[n-i-j, k]], {j, k, n}, {i, k, n-j}]];
%t A181366 T[n_, k_] := A[n, k] - A[n, k+1];
%t A181366 a[n_] := Sum[k T[n, k], {k, 0, terms}];
%t A181366 Array[a, terms] (* _Jean-François Alcover_, Nov 11 2020, after _Alois P. Heinz_ in A181365 *)
%Y A181366 Cf. A181365.
%K A181366 nonn
%O A181366 1,3
%A A181366 _Emeric Deutsch_, Oct 15 2010