A058300 Number of ways of piling up n wine bottles above a row of n+1 bottles at ground level.

1, 1, 1, 3, 7, 16, 43, 115, 303, 813, 2203, 5991, 16371, 44917, 123598, 340988, 942930, 2612735, 7252407, 20163046, 56136326, 156488946, 436739752, 1220157514, 3412116339, 9550192161, 26751643663, 74991516850, 210364915858, 590490257667, 1658484275955

Offset

0,4

Comments

Related to the Catalan numbers (which count the ways of storing an arbitrary number of bottles above n bottles at ground level).

Related to fountains of n coins (A005169). [Joerg Arndt, Mar 18 2011]

References

R. P. Stanley, Enumerative Combinatorics (Volume 2); see Exercise 6.19(hhh).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..600

Andrew M. Odlyzko and Herbert S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Formula

Coefficient of w^(2*n+1)*z^(n+1) in the formal power series G(w, z) defined by G(w, z)=1+w*z*G(w, w*z).

a(n) = A047998(2n+1,n+1). - Alois P. Heinz, Jun 24 2015

a(n) ~ c * d^n / sqrt(n), where d = 2.8566122635122125634030051... and c = 0.19212135026441477122126... - Vaclav Kotesovec, Jul 17 2019

Example

a(4) = 7: the seven possibilities are:
..............0.............0.........0...............0.........0............0
.0.0.0.0.....0.0.0.......0.0.0.......0.0...0.....0...0.0.......0.0.0......0.0.0
0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0.,.0.0.0.0.0,.0.0.0.0.0

Mathematica

terms = 31; initialMax = 5; Clear[g]; g[max_] := g[max] = (Print["max = ", max]; f = 1/Fold[1 - y*x^#2/#1&, 1, Range[max] // Reverse]; b[n_, k_] := SeriesCoefficient[f, {x, 0, n}, {y, 0, k}]; b[0, 0] = 1; Clear[a]; a[n_] := a[n] = b[2n+1, n+1]; Array[a, terms, 0]); g[max = initialMax]; g[max = max+1]; While[g[max] != g[max-1], max = max+1]; A058300 = g[max] (* Jean-François Alcover, Oct 05 2017, after Alois P. Heinz's formula *)

Crossrefs

Cf. A047998.

Sequence in context: A036567 A018023 A144977 * A000674 A129045 A217358

Adjacent sequences: A058297 A058298 A058299 * A058301 A058302 A058303

Keyword

easy,nonn,nice

Author

Roland Bacher, Dec 08 2000

Extensions

More terms from Alois P. Heinz, Jun 24 2015

Status

approved