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

1,1

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) = 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

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

Prog

(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

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