A014605 -id:A014605 - cp's OEIS frontend

A007478 Dimension of primitive Vassiliev knot invariants of order n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55

Offset: 0

N. J. A. Sloane

Next term is at least 78. - Jan Kneissler jk(AT)math.uni-bonn.de, Sep 1997

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
D. Bar-Natan, Bibliography of Vassiliev Invariants.
Joan S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253-287; TeX source.
D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants, arXiv:q-alg/9709031, 1997.
S. Chmutov and S. Duzhin, A lower bound for the number of Vassiliev knot invariants, Topology and its Applications, Volume 92, Number 3, 14 April 1999, pp. 201-223(23).
Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12, arXiv:q-alg/9706022, 1997.
T. Ohtsuki (ed.), Problems on invariants of knots and 3-manifolds, arXiv:math/0406190 [math.GT], (2004); see Table 2 on p.407.
Index entries for sequences related to knots

Cf. A007473, A014605.

Broadhurst gives a conjectured g.f.

A014595 Conjectured dimensions of spaces of weight systems of chord diagrams.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 33, 60, 104, 184, 316, 548, 931, 1588, 2676, 4511, 7539, 12590, 20890, 34603, 57036, 93804, 153655, 251109, 408961, 664467, 1076398, 1739660, 2804166, 4510035, 7236242, 11585908, 18509442, 29511312, 46957178, 74575323, 118213424, 187052097, 295453415

Offset: 0

David Broadhurst

nonn

First 13 terms agree with A007473, next terms obtained from A014605.

D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
D. Bar-Natan, Bibliography of Vassiliev Invariants.
Birman, Joan S., New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253-287.
D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants, arXiv:q-alg/9709031, 1997 [different conjectured values].
Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12, arXiv:q-alg/9706022, 1997.
Index entries for sequences related to knots

Cf. A014605 (conjectured to agree with A007478), A014596 (first differences, conjectured to agree with A007293). This sequences is conjectured to agree with A007473. All these (equivalent) conjectures are probably wrong since Jan Kneissler states that A007478(13) >= 78 (see A007478), while A014605(13)=77.

terms = 40; QP = QPochhammer; A014605 = Join[{1, 0, 0, 0}, CoefficientList[ QP[q^4]/QP[q] + O[q]^terms, q]] // Accumulate;
gf = Product[(1 - y^m)^(-A014605[[m+1]]), {m, 1, terms}] + O[y]^terms;
CoefficientList[gf, y] (* Jean-François Alcover, Jul 21 2018 *)

G.f.: Product_{ m >= 1 } (1-y^m)^(-A014605(m)). - Andrey Zabolotskiy, Sep 15 2017

More terms from Joerg Arndt, Sep 19 2017

cp's OEIS Frontend

A007478 Dimension of primitive Vassiliev knot invariants of order n.

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