A120420 Number of graphs of functions with n triple points.

1, 2, 19, 428, 17746, 1178792, 114892114, 15465685088, 2750970320776, 625218940868432, 176816078759177584, 60910324282361987648, 25113805147553980835056, 12212353714587611449244672

Offset

1,2

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), 111-118.

Teena Carroll, David Galvin, The game of plates and olives, arXiv:1711.10670 [math.CO], 2017.

Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005-2006.

Formula

Nicolaescu gives a g.f.

a(n) >= A135487(n).

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 = k-y; If[y==0, A[x, y] = 1/(2^x), If[x>0, A[x, y] = (1/(x+2y+1))((x+1)A[x+1, y-1] + 1/2 (x+1)A[x-1, y] + 1/2 (x+1)Sum[Sum[A[i, j]A[x-i, y-1-j], {j, 0, y-1}], {i, 0, x}]), A[x, y] = (1/(2y+1))((x+1)A[x+1, y-1] + 1/2 (x+1)Sum[Sum[A[i, j]A[x-i, y-1-j], {j, 0, y-1}], {i, 0, x}])]]]]; a! (2b+1)! A[a, b]];
Table[Morse[0, n], {n, 0, 13}] (* Jean-François Alcover, Oct 05 2018, after Nicolaescu's code *)

Crossrefs

Sequence in context: A379897 A353290 A332967 * A350939 A239674 A306457

Adjacent sequences: A120417 A120418 A120419 * A120421 A120422 A120423

Keyword

nonn

Author

N. J. A. Sloane, Feb 21 2008

Extensions

More terms from Vladeta Jovovic, Feb 22 2008

Status

approved