A008985 - OEIS

%I #27 Jan 22 2025 01:59:56

%S 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,

%T 290,133,26,80,538,1504,2370,2370,1504,538,80,246,2144,7607,14846,

%U 18273,14846,7607,2144,246,810,8643,37762,90182,134855,134855,90182,37762,8643,810

%N 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.

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

%H Andrey Zabolotskiy, <a href="/A008985/b008985.txt">Table of n, a(n) for n = -1..89</a> (rows -1..11)

%H S. V. Duzhin, <a href="https://www.pdmi.ras.ru/~arnsem/dataprog/">Computer programs and data files of the participants of Arnold's seminar</a>, 1998-2010.

%H S. M. Gusein-Zade and F. S. Duzhin, <a href="https://doi.org/10.4213/rm48">On the number of topological types of plane curves</a>; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. <a href="https://doi.org/10.1070/rm1998v053n03abeh000048">English translation</a>: Russian Mathematical Surveys 53 (1998) 626-627.

%H Andrey Zabolotskiy, <a href="https://github.com/colt-browning/closed_curves">closed_curves</a> - a program computing the triangle (2025).

%F T(n, n+1) = A003239(n). [Arnold, p. 12] - _Andrey Zabolotskiy_, Oct 22 2023

%e Triangle begins:

%e   0;

%e   1, 1;

%e   1, 1, 1;

%e   2, 3, 3, 2;

%e   4, 10, 11, 10, 4;

%e   10, 35, 57, 57, 35, 10;

%e   ...

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

%K nonn,nice,tabl

%O -1,7

%A _N. J. A. Sloane_

%E Name clarified and initial row added by _Andrey Zabolotskiy_, Oct 22 2023

%E Rows 6-8 from _Andrey Zabolotskiy_, Jan 19 2025