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, 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

Andrew Howroyd, Table of n, a(n) for n = 0..50

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

Adjacent sequences: A295926 A295927 A295928 * A295930 A295931 A295932

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