
LINKS

Vladeta Jovovic, Feb 22 2008, Table of n, a(n) for n = 1..21
V. I. Arnold, Smooth function statistics, Funct. Anal. Other. Math., 1 (2006), 111118.
Teena Carroll, David Galvin, The game of plates and olives, arXiv:1711.10670 [math.CO], 2017.
Liviu I. Nicolaescu, Counting Morse functions on the 2sphere, arXiv:math/0512496 [math.GT], 20052006.


MATHEMATICA

Morse[a_, b_] := Module[{i, j, k, m, x, y, A}, A[0, 0]=1; m = a+b; For[k=1, k <= m, k++, For[y=0, y <= b, y++, x = ky; If[y==0, A[x, y] = 1/(2^x), If[x>0, A[x, y] = (1/(x+2y+1))((x+1)A[x+1, y1] + 1/2 (x+1)A[x1, y] + 1/2 (x+1)Sum[Sum[A[i, j]A[xi, y1j], {j, 0, y1}], {i, 0, x}]), A[x, y] = (1/(2y+1))((x+1)A[x+1, y1] + 1/2 (x+1)Sum[Sum[A[i, j]A[xi, y1j], {j, 0, y1}], {i, 0, x}])]]]]; a! (2b+1)! A[a, b]];
Table[Morse[0, n], {n, 0, 13}] (* JeanFrançois Alcover, Oct 05 2018, after Nicolaescu's code *)
