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A181306
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Number of 2-compositions of n having no increasing columns. 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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2
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1, 1, 3, 7, 18, 44, 110, 272, 676, 1676, 4160, 10320, 25608, 63536, 157648, 391152, 970528, 2408064, 5974880, 14824832, 36783296, 91266496, 226449920, 561866240, 1394099328, 3459031296, 8582528768, 21294921472, 52836837888, 131098461184
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OFFSET
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0,3
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COMMENTS
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Also, number of 2-compositions of n that have no odd entries in the top row. 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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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. =(1+z)(1-z)^2/(1-2z-2z^2+2z^3).
a(0)=1, a(1)=1, a(2)=3, a(3)=7, a(n)=2*a(n-1)+2*a(n-2)-2*a(n-3) [From Harvey P. Dale, Mar 07 2012]
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EXAMPLE
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a(2)=3 because we have (1/1), (2/0), and (1,1/0,0) (the 2-compositions are written as (top row / bottom row).
Alternatively, a(2)=3 because we have (0/2),(2,0), and (0,0/1,1) (the 2-compositions are written as (top row/bottom row)).
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MAPLE
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g := (1+z)*(1-z)^2/(1-2*z-2*z^2+2*z^3): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
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MATHEMATICA
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CoefficientList[Series[((1+x)(1-x)^2)/(1-2x-2x^2+2x^3), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{2, 2, -2}, {1, 3, 7}, 30]] (* Harvey P. Dale, Mar 07 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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