Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #44 May 29 2018 20:37:33
%S 1,2,10,76,772,9856,152099,2758931,57602672,1362342830,36046013013,
%T 1056342305565,34002625115587,1193660155852584,45414253886783716,
%U 1862232981974586960,81893921416048297995,3845201559359081046971,192006280895048080286802
%N a(n) is the number of topological equivalence classes of excellent Morse functions on S^2 with 2n+2 critical points (n saddle points).
%C 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.
%D L. Nicolaescu, Counting Morse functions on the 2-sphere, Compositio Math. 144.
%H Andrew Howroyd, <a href="/A295929/b295929.txt">Table of n, a(n) for n = 0..50</a>
%H Teena Carroll, David Galvin, <a href="https://arxiv.org/abs/1711.10670">The game of plates and olives</a>, arXiv:1711.10670 [math.CO], 2017.
%H Andrew Howroyd, <a href="/A295929/a295929.txt">PARI Code</a>
%H L. Nicolaescu, <a href="https://www3.nd.edu/~lnicolae/MorseCountCompv3.pdf">Counting Morse functions on the 2-sphere</a>, Compositio Math. 144 (2008).
%F a(n) >= A001147(n) = (2*n - 1)!!. - _David A. Corneth_, Nov 30 2017
%e From _David A. Corneth_, Nov 30 2017: (Start)
%e a(0) = 1 as there is exactly one way to get an empty table for the first time in two steps:
%e Step 1: an empty plate is placed on the table.
%e Step 2: an empty plate is removed from the table. (End)
%Y Cf. A001147.
%K nonn
%O 0,2
%A _Michel Marcus_, Nov 30 2017
%E a(6)-a(18) from _Kyle Weingartner_, Dec 04 2017
%E New name from _Kyle Weingartner_, Dec 05 2017