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%I #2 Mar 30 2012 17:36:24
%S 2,5,2,12,10,2,29,41,10,2,70,152,46,10,2,169,536,193,46,10,2,408,1830,
%T 770,198,46,10,2,985,6120,2972,811,198,46,10,2,2378,20178,11202,3218,
%U 816,198,46,10,2,5741,65867,41481,12484,3259,816,198,46,10,2,13860
%N Triangle read by rows: T(n,k) is the number of 2-compositions of n having largest entry k (1<=k<=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 The sum of entries in row n is A003480(n).
%C T(n,1)=A000129(n+1) (the Pell numbers).
%C Sum(k*T(n,k),k=0..n)=A181339.
%D G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
%F G.f. for 2-compositions with all entries <= k is h(k,z)=(1-z)^2/(1-4z+2z^2+2z^{k+1}-z^{2k+2}).
%F 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 G.f. = G(t,z)=Sum(f(k,z)*t^k, k=1..infinity).
%e T(3,3)=2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row).
%e Triangle starts:
%e 2;
%e 5,2;
%e 12,10,2;
%e 29,41,10,2;
%e 70,152,46,10,2;
%p 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 Cf. A003480, A000129, A181339
%K nonn,tabl
%O 1,1
%A _Emeric Deutsch_, Oct 15 2010
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