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

3,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots

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

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 *)

Crossrefs

Cf. A006134.

Sequence in context: A082819 A298733 A197311 * A346109 A362220 A019778

Adjacent sequences: A078475 A078476 A078477 * A078479 A078480 A078481

Keyword

nonn

Author

Ralf Stephan, Jan 03 2003

Extensions

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

Status

approved