A008985 Triangle T(n,k) giving number of immersions of the oriented circle into the oriented plane with n double points and index k, k = -n-1, -n+1, ..., n-1, n+1.

0, 1, 1, 1, 1, 1, 2, 3, 3, 2, 4, 10, 11, 10, 4, 10, 35, 57, 57, 35, 10, 26, 133, 290, 364, 290, 133, 26, 80, 538, 1504, 2370, 2370, 1504, 538, 80, 246, 2144, 7607, 14846, 18273, 14846, 7607, 2144, 246, 810, 8643, 37762, 90182, 134855, 134855, 90182, 37762, 8643, 810

Offset

-1,7

References

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.

Links

Andrey Zabolotskiy, Table of n, a(n) for n = -1..89 (rows -1..11)

S. V. Duzhin, Computer programs and data files of the participants of Arnold's seminar, 1998-2010.

S. M. Gusein-Zade and F. S. Duzhin, On the number of topological types of plane curves; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. English translation: Russian Mathematical Surveys 53 (1998) 626-627.

Andrey Zabolotskiy, closed_curves - a program computing the triangle (2025).

Formula

T(n, n+1) = A003239(n). [Arnold, p. 12] - Andrey Zabolotskiy, Oct 22 2023

Example

Triangle begins:
  0;
  1, 1;
  1, 1, 1;
  2, 3, 3, 2;
  4, 10, 11, 10, 4;
  10, 35, 57, 57, 35, 10;
  ...

Crossrefs

Cf. A008980 (row sums), A008981, A008982, A008983, A054993, A008984 (k = n-1), A003239 (k = n+1).

Sequence in context: A073078 A360296 A034799 * A326699 A138652 A323833

Adjacent sequences: A008982 A008983 A008984 * A008986 A008987 A008988

Keyword

nonn,nice,tabl

Author

N. J. A. Sloane

Extensions

Name clarified and initial row added by Andrey Zabolotskiy, Oct 22 2023

Rows 6-8 from Andrey Zabolotskiy, Jan 19 2025

Status

approved