|
%I #28 Apr 17 2023 15:52:30
%S 2,16,208,3584,76544,1947648,57477120,1929117696,72545402880,
%T 3020819005440,137959904378880,6855868809216000,368270708268072960,
%U 21262037565623500800,1312956239068318924800,86347473137975269785600,6025205587810776514560000,444600907757468888806195200
%N Number of ways of arranging the numbers 1 ... 2n into a 2 X n array so there is exactly one local maximum.
%C Also the number of random walk labelings of the grid graph P_2 X P_n. - _Sela Fried_, Apr 14 2023
%H Andrew Howroyd, <a href="/A087923/b087923.txt">Table of n, a(n) for n = 1..200</a>
%H Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2304.05728">Graph labelings obtainable by random walks</a>, arXiv:2304.05728 [math.CO], 2023.
%F 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
%F From _Sela Fried_, Apr 13 2023: (Start)
%F 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).
%F 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).
%F (End)
%F a(n) ~ Pi * 2^(2*n - 5/2) * n^(n+1) / exp(n). - _Vaclav Kotesovec_, Apr 13 2023
%p 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):
%p seq(a(n), n = 1..18); # _Peter Luschny_, Apr 17 2023
%o (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
%Y Row 2 of A087783.
%Y Cf. A007846.
%K nonn
%O 1,1
%A _R. H. Hardin_, Oct 27 2003
%E Terms a(16) and beyond from _Andrew Howroyd_, Feb 26 2020
| 