A261683 Number of permutations p of {1..2n} such that p[2k-1]<p[2k] <=> p[2k]<p[2k+1].

2, 8, 84, 1632, 51040, 2340480, 147985824, 12338740736, 1311694023168, 173163464017920, 27793189979315200, 5329882370469617664, 1203569385876087300096, 316106247473967737765888, 95541594110304706706657280, 32926404311225961897742172160

Offset

1,1

Comments

The relation between p[2n-1] and p[2n] is arbitrary; hence a(n) = 2*n*A122647(n). a(n) is also (surprisingly) the number of 2 X n whirlpool permutations (see link, also A334518). - Don Knuth, May 06 2020.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..245

Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.

D. E. Knuth, Whirlpool Permutations, May 05 2020.

Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.

Formula

Basset (2014, Eq. (4)) gives a g.f.

a(n) = (2n)! [z^(2n)] 2*sqrt(2)*z*(exp(sqrt(2)*z)-1) / (2+sqrt(2)*z + (2-sqrt(2)*z)*exp(sqrt(2)*z)). - Alois P. Heinz, Sep 06 2015

Maple

egf:= 2*(x->1/(1-x*tanh(x))-1)(z/sqrt(2)):
a:= n-> (2*n)!*coeff(series(egf, z, 2*n+1), z, 2*n):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 06 2015

Crossrefs

Cf. A122647, A334518, A334519.

Sequence in context: A336252 A276488 A295764 * A134089 A136647 A306001

Adjacent sequences: A261680 A261681 A261682 * A261684 A261685 A261686

Keyword

nonn

Author

N. J. A. Sloane, Sep 05 2015

Extensions

More terms from Alois P. Heinz, Sep 06 2015

Name corrected by Don Knuth. - N. J. A. Sloane, May 06 2020

Status

approved