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A295929
a(n) is the number of topological equivalence classes of excellent Morse functions on S^2 with 2n+2 critical points (n saddle points).
1
1, 2, 10, 76, 772, 9856, 152099, 2758931, 57602672, 1362342830, 36046013013, 1056342305565, 34002625115587, 1193660155852584, 45414253886783716, 1862232981974586960, 81893921416048297995, 3845201559359081046971, 192006280895048080286802
OFFSET
0,2
COMMENTS
a(n) is also the number of ways of returning to an empty table for the first time after exactly 2n + 2 steps in the game of plates and olives. See the Carroll & Galvin link for a description of the game of plates and olives.
REFERENCES
L. Nicolaescu, Counting Morse functions on the 2-sphere, Compositio Math. 144.
LINKS
Teena Carroll, David Galvin, The game of plates and olives, arXiv:1711.10670 [math.CO], 2017.
Andrew Howroyd, PARI Code
L. Nicolaescu, Counting Morse functions on the 2-sphere, Compositio Math. 144 (2008).
FORMULA
a(n) >= A001147(n) = (2*n - 1)!!. - David A. Corneth, Nov 30 2017
EXAMPLE
From David A. Corneth, Nov 30 2017: (Start)
a(0) = 1 as there is exactly one way to get an empty table for the first time in two steps:
Step 1: an empty plate is placed on the table.
Step 2: an empty plate is removed from the table. (End)
CROSSREFS
Cf. A001147.
Sequence in context: A066223 A355110 A088500 * A195136 A294573 A301741
KEYWORD
nonn
AUTHOR
Michel Marcus, Nov 30 2017
EXTENSIONS
a(6)-a(18) from Kyle Weingartner, Dec 04 2017
New name from Kyle Weingartner, Dec 05 2017
STATUS
approved