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A087923
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Number of ways of arranging the numbers 1 ... 2n into a 2 X n array so there is exactly one local maximum.
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2
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2, 16, 208, 3584, 76544, 1947648, 57477120, 1929117696, 72545402880, 3020819005440, 137959904378880, 6855868809216000, 368270708268072960, 21262037565623500800, 1312956239068318924800, 86347473137975269785600, 6025205587810776514560000, 444600907757468888806195200
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OFFSET
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1,1
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COMMENTS
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Also the number of random walk labelings of the grid graph P_2 X P_n. - Sela Fried, Apr 14 2023
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LINKS
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FORMULA
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a(n) = 2*Sum_{k=1..n} (2*n-2)!*(2*k*(n-k+1)-1)/((2*k-1)!!*(2*n-2*k-1)!!). - Maximilian Göbel, Feb 26 2020
a(n) = 2^(n - 1)*(n - 1)!*Sum_{k=0..n-1} (n*binomial(2*(n - 1), 2*k) + binomial(2*n - 1, 2*k))/binomial(n - 1,k).
E.g.f.: ((1 - 2*x)^2*arctan(2*x/sqrt(1 - 4*x)) + 2*x*sqrt(1 - 4*x))/(2*(sqrt(1 - 4*x))^3).
(End)
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MAPLE
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a := n -> 2*((2*n - 2)! / doublefactorial(2*n - 1)) * add((2*k*(n - k + 1) - 1) * binomial(2*n, 2*k) / binomial(n, k), k = 1..n):
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PROG
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(PARI) a(n)={2*sum(k=1, n, (2*n-2)!*(2*k*(n-k+1)-1)*2^n*k!*(n-k)!/((2*k)!*(2*n-2*k)!))} \\ Andrew Howroyd, Feb 26 2020
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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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