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A058300
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Number of ways of piling up n wine bottles above a row of n+1 bottles at ground level.
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2
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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
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OFFSET
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0,4
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COMMENTS
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Related to the Catalan numbers (which count the ways of storing an arbitrary number of bottles above n bottles at ground level).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics (Volume 2); see Exercise 6.19(hhh).
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LINKS
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FORMULA
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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) ~ c * d^n / sqrt(n), where d = 2.8566122635122125634030051... and c = 0.19212135026441477122126... - Vaclav Kotesovec, Jul 17 2019
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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