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 A181299 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)). 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. 2

%I

%S 1,2,6,1,20,4,64,17,1,206,68,6,662,261,32,1,2128,976,152,8,6840,3577,

%T 675,51,1,21986,12912,2860,280,10,70670,46049,11704,1406,74,1,227156,

%U 162628,46632,6632,460,12,730152,569705,181877,29866,2570,101,1,2346942

%N Triangle read by rows: T(n,k) is the number of 2-compositions of n having k columns in which the top entry is equal to the bottom entry (0<=k<=floor(n/2)). 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 Row n contains 1+floor(n/2) entries.

%C The sum of entries in row n is A003480(n).

%C T(n,0)=A181301(n).

%C Sum(k*T(n,k),k>=0)=A181300.

%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.=G(t,z)=(1+z)(1-z)^2/[(1-3z-z^2+z^3-t(1-z)z^2].

%e T(3,1)=4 because we have (1,1/1,0),(1,0/1,1),(1,1/0,1),(0,1/1,1) (the 2-compositions are written as (top row/bottom row).

%e Triangle starts:

%e 1;

%e 2;

%e 6,1;

%e 20,4;

%e 64,17,1;

%p G := (1-z)^2*(1+z)/(1-3*z-z^2+z^3-t*z^2*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A003480, A181300, A181301.

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, Oct 12 2010

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