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%I #24 Aug 16 2020 02:33:54
%S 1,1,1,3,7,16,43,115,303,813,2203,5991,16371,44917,123598,340988,
%T 942930,2612735,7252407,20163046,56136326,156488946,436739752,
%U 1220157514,3412116339,9550192161,26751643663,74991516850,210364915858,590490257667,1658484275955
%N Number of ways of piling up n wine bottles above a row of n+1 bottles at ground level.
%C Related to the Catalan numbers (which count the ways of storing an arbitrary number of bottles above n bottles at ground level).
%C Related to fountains of n coins (A005169). [_Joerg Arndt_, Mar 18 2011]
%D R. P. Stanley, Enumerative Combinatorics (Volume 2); see Exercise 6.19(hhh).
%H Alois P. Heinz, <a href="/A058300/b058300.txt">Table of n, a(n) for n = 0..600</a>
%H Andrew M. Odlyzko and Herbert S. Wilf, <a href="http://www.jstor.org/stable/2322898">The editor's corner: n coins in a fountain</a>, Amer. Math. Monthly, 95 (1988), 840-843.
%F 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).
%F a(n) = A047998(2n+1,n+1). - _Alois P. Heinz_, Jun 24 2015
%F a(n) ~ c * d^n / sqrt(n), where d = 2.8566122635122125634030051... and c = 0.19212135026441477122126... - _Vaclav Kotesovec_, Jul 17 2019
%e a(4) = 7: the seven possibilities are:
%e ..............0.............0.........0...............0.........0............0
%e .0.0.0.0.....0.0.0.......0.0.0.......0.0...0.....0...0.0.......0.0.0......0.0.0
%e 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
%t 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 *)
%Y Cf. A047998.
%K easy,nonn,nice
%O 0,4
%A _Roland Bacher_, Dec 08 2000
%E More terms from _Alois P. Heinz_, Jun 24 2015
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