A087923 - OEIS

A087923

Number of ways of arranging the numbers 1 ... 2n into a 2 X n array so there is exactly one local maximum.

2, 16, 208, 3584, 76544, 1947648, 57477120, 1929117696, 72545402880, 3020819005440, 137959904378880, 6855868809216000, 368270708268072960, 21262037565623500800, 1312956239068318924800, 86347473137975269785600, 6025205587810776514560000, 444600907757468888806195200 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`Also the number of random walk labelings of the grid graph P_2 X P_n. - Sela Fried, Apr 14 2023`
	LINKS	`Andrew Howroyd, Table of n, a(n) for n = 1..200` `Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.`
	FORMULA	`a(n) = 2Sum_{k=1..n} (2n-2)!(2k(n-k+1)-1)/((2k-1)!!(2n-2k-1)!!). - Maximilian Göbel, Feb 26 2020` `From Sela Fried, Apr 13 2023: (Start)` `a(n) = 2^(n - 1)(n - 1)!Sum_{k=0..n-1} (nbinomial(2(n - 1), 2k) + binomial(2n - 1, 2k))/binomial(n - 1,k).` `E.g.f.: ((1 - 2x)^2arctan(2x/sqrt(1 - 4x)) + 2xsqrt(1 - 4x))/(2(sqrt(1 - 4x))^3).` `(End)` `a(n) ~ Pi 2^(2n - 5/2) n^(n+1) / exp(n). - Vaclav Kotesovec, Apr 13 2023`
	MAPLE	`a := n -> 2((2n - 2)! / doublefactorial(2n - 1)) add((2k(n - k + 1) - 1) * binomial(2n, 2k) / binomial(n, k), k = 1..n):` `seq(a(n), n = 1..18); # Peter Luschny, Apr 17 2023`
	PROG	`(PARI) a(n)={2sum(k=1, n, (2n-2)!(2k(n-k+1)-1)2^nk!(n-k)!/((2k)!(2n-2k)!))} \\ Andrew Howroyd, Feb 26 2020`
	CROSSREFS	`Row 2 of A087783.` `Cf. A007846.` `Sequence in context: A360466 A161568 A138429 * A222523 A004121 A188600` `Adjacent sequences: A087920 A087921 A087922 * A087924 A087925 A087926`
	KEYWORD	`nonn`
	AUTHOR	`R. H. Hardin, Oct 27 2003`
	EXTENSIONS	`Terms a(16) and beyond from Andrew Howroyd, Feb 26 2020`
	STATUS	`approved`