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A225006
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Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.
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3
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1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080
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OFFSET
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0,2
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COMMENTS
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Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013
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LINKS
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FORMULA
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Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).
a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015
It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
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EXAMPLE
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Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01: [ 1 1 ]
02: [ 1 2 ]
03: [ 1 3 ]
04: [ 2 1 ]
05: [ 2 2 ]
06: [ 2 3 ]
07: [ 3 1 ]
08: [ 3 2 ]
09: [ 3 3 ]
(End)
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MATHEMATICA
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PROG
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(PARI) { a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }
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CROSSREFS
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Cf. A088536 (unimodal maps [1..n]->[1..n]).
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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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