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A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=n).

%I A181338 #8 May 11 2025 09:08:12
%S A181338 2,5,2,12,10,2,29,41,10,2,70,152,46,10,2,169,536,193,46,10,2,408,1830,
%T A181338 770,198,46,10,2,985,6120,2972,811,198,46,10,2,2378,20178,11202,3218,
%U A181338 816,198,46,10,2,5741,65867,41481,12484,3259,816,198,46,10,2,13860
%N A181338 Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=n).
%C A181338 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 A181338 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.
%F A181338 G.f. for 2-compositions with all entries <= k is h(k,z) = (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)).
%F A181338 G.f. for 2-compositions with largest entry k is f(k,z) = h(k,z)-h(k-1,z) (these are the column g.f.'s).
%F A181338 G.f.: G(t,z) = Sum_{k>=1} f(k,z)*t^k.
%F A181338 Sum_{k=0..n} T(n,k) = A003480(n).
%F A181338 T(n,1) = A000129(n+1) (the Pell numbers).
%F A181338 Sum_{k=0..n} k*T(n,k) = A181339(n).
%e A181338 T(3,3) = 2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row)).
%e A181338 Triangle starts:
%e A181338   2;
%e A181338   5,2;
%e A181338   12,10,2;
%e A181338   29,41,10,2;
%e A181338   70,152,46,10,2;
%p A181338 h := proc (k) options operator, arrow: (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)-h(k-1)) end proc: G := sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 11 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
%Y A181338 Cf. A003480, A000129, A181339
%K A181338 nonn,tabl
%O A181338 1,1
%A A181338 _Emeric Deutsch_, Oct 15 2010