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A181338
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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.
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1
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2, 5, 2, 12, 10, 2, 29, 41, 10, 2, 70, 152, 46, 10, 2, 169, 536, 193, 46, 10, 2, 408, 1830, 770, 198, 46, 10, 2, 985, 6120, 2972, 811, 198, 46, 10, 2, 2378, 20178, 11202, 3218, 816, 198, 46, 10, 2, 5741, 65867, 41481, 12484, 3259, 816, 198, 46, 10, 2, 13860
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OFFSET
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1,1
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COMMENTS
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The sum of entries in row n is A003480(n).
T(n,1)=A000129(n+1) (the Pell numbers).
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REFERENCES
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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.
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LINKS
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FORMULA
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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}).
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).
G.f. = G(t,z)=Sum(f(k,z)*t^k, k=1..infinity).
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EXAMPLE
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T(3,3)=2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row).
Triangle starts:
2;
5,2;
12,10,2;
29,41,10,2;
70,152,46,10,2;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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