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

Offset: 1

Author

R. H. Hardin, Oct 27 2003

Keywords

• nonn

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.

Crossrefs

Row 2 of A087783.

Cf. A007846.

Programs

Maple

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):
seq(a(n), n = 1..18); # Peter Luschny, Apr 17 2023

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

Formula

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

From Sela Fried, Apr 13 2023: (Start)

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)

a(n) ~ Pi * 2^(2*n - 5/2) * n^(n+1) / exp(n). - Vaclav Kotesovec, Apr 13 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Feb 26 2020