A078478 Number of rational knots of n crossings with signature 0 (chiral pairs counted twice).

0, 1, 0, 3, 2, 9, 6, 29, 30, 99, 112, 351, 450, 1275, 1734, 4707, 6762, 17577, 26208, 66197, 101862, 250953, 395804, 956385, 1540110, 3660541, 5997600, 14061141, 23382294, 54177741, 91246662, 209295261, 356432166, 810375651, 1393592512

Offset: 3

Author

Ralf Stephan, Jan 03 2003

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots

Crossrefs

Cf. A006134.

Programs

Mathematica

CoefficientList[Series[(- x/2) (2 + (2*x^4 - x^2 - 1) / (Sqrt[1 - 4 x^4] (1 + x^2)) + (2 x^2 - x - 1) / (Sqrt[1 - 4 x^2] (1 + x))) / x^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 17 2013 *)

Formula

G.f.: (-x/2)*( 2 + (2*x^4-x^2-1)/(sqrt(1-4*x^4)*(1+x^2)) + (2*x^2-x-1)/(sqrt(1-4*x^2)*(1+x)) ) - Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006

a(2n+4) = A006134(n-1) = Sum[ (2k)!/(k!)^2, {k,0,n} ]. - Alexander Adamchuk, Feb 23 2007

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006