A322634 Sum of attendance numbers of all histories of length 5*n in the Bizley-Duchon's club model, divided by 5.

5, 153, 4537, 133189, 3891675, 113415423, 3299905647

Offset

1,1

Comments

The Bizley-Duchon's club model is equivalent to the lattice paths from (0,0) to (3*n,2*n) described in A293946. The attendance history of the club consists of persons entering in pairs and leaving in groups of three. The club closes when no persons are remaining. a(k)/A293946(k) is proportional to the mean area under the "filling level curve" of the club.

Banderier et al. show that the mean area is asymptotic to K*n^(3/2), with K=(1/2)*(15*Pi)^(1/2).

Links

Table of n, a(n) for n=1..7.

Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area. Chassaing, Philippe and others. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.345-356, 2006, DMTCS Proceedings.

Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.

Example

a(1) = (15 + 10)/5 = 5:
  Contributions of the A293946(1) = 2 attendance histories are
  0 (+2) 2 (+2) 4 (+2) 6 (-3) 3 (-3) 0 -> 2 + 4 + 6 + 3 = 15
  0 (+2) 2 (+2) 4 (-3) 1 (+2) 3 (-3) 0 -> 2 + 4 + 1 + 3 = 10.

Crossrefs

Cf. A060941, A293946.

Sequence in context: A105230 A222768 A208578 * A108535 A265123 A237527

Adjacent sequences: A322631 A322632 A322633 * A322635 A322636 A322637

Keyword

nonn,more

Author

Hugo Pfoertner, Dec 22 2018

Status

approved