A047051 Prime alternating tangle types (of knots) with n crossings.

1, 2, 4, 10, 29, 98, 372, 1538, 6755, 30996, 146982, 715120, 3552254, 17951322, 92045058, 477882876, 2508122859, 13289437362, 71010166670, 382291606570, 2072025828101, 11298920776704, 61954857579594, 341427364138880

Offset

1,2

Comments

G.f. is related to the classes of 2- and 3-connected planar maps with n edges. Further terms are known.

References

C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.

Links

Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007, Table of n, a(n) for n = 1..50

S. R. Finch, Knots, links and tangles

S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]

C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, p. 12.

P. Zinn-Justin, The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

Mathematica

max = 24; Clear[a, eq, s]; gf = Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 2; coes = CoefficientList[(x^4 - 2*x^3 + x^2)*gf^5 + (8*x^4 - 14*x^3 + 8*x^2 - 2*x)*gf^4 + (25*x^4 - 16*x^3 - 14*x^2 + 8*x + 1)*gf^3 + (38*x^4 + 15*x^3 - 30*x^2 - x + 2)*gf^2 + (28*x^4 + 36*x^3 - 5*x^2 - 12*x + 1)*gf + 8*x^4 + 17*x^3 + 8*x^2 - x, x]; eq[n_] := eq[n] = If[n == 1, Thread[Drop[coes, 3] == 0], eq[n-1] /. s[n-1] // First]; s[n_] := s[n] = (Print["n = ", n]; Solve[eq[n][[n]], a[n+2]]); sol = Table[s[n], {n, 1, max-2}] // Flatten; Table[a[n], {n, 1, max}] /. sol (* Jean-François Alcover, Apr 15 2014 *)

Crossrefs

Cf. A002864, A000139, A000287.

Sequence in context: A230957 A279552 A261041 * A271077 A365903 A126349

Adjacent sequences: A047048 A047049 A047050 * A047052 A047053 A047054

Keyword

easy,nice,nonn

Author

Valery A. Liskovets

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

Status

approved