A007478 -id:A007478 - cp's OEIS frontend

A014605 Partial sums of A001935; at one time this was conjectured to agree with A007478.

1, 1, 1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 77, 106, 144, 194, 258, 340, 445, 577, 743, 951, 1209, 1529, 1924, 2408, 3000, 3722, 4598, 5658, 6938, 8477, 10323, 12533, 15169, 18307, 22035, 26451, 31673, 37836, 45092, 53620, 63626, 75342, 89038, 105024, 123648

Offset: 0

David Broadhurst

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Bar-Natan, Dror, 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.
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

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(`if`(
      irem(d, 4)=0, 0, d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<4, 1, a(n-1)+b(n-4)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 21 2018

QP = QPochhammer; Join[{1, 0, 0, 0}, CoefficientList[QP[q^4]/QP[q]+O[q]^50, q]] // Accumulate (* Jean-François Alcover, Jul 21 2018 *)

a(n) = a(n-1) + A001935(n-4), n>3. - R. J. Mathar, Mar 06 2016

A007293 Dimension of space of weight systems of chord diagrams.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 27, 44, 80, 132, 232

Offset: 0

N. J. A. Sloane

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.
J. S. Birman, Letter to N. J. A. Sloane, Apr 09 1994
J. S. Birman, New points of view in knot theory.. Bulletin of the American Mathematical Society 28.2 (1993): 253-287.
D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants.
Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12
T. Ohtsuki (ed.), Problems on invariants of knots and 3-manifolds, arXiv:math/0406190 [math.GT], (2004); see Table 3 on p.408.
Evert Stenlund, On the Vassiliev Invariants, June 2017.
S. D. Tyurina, Diagram invariants of knots and the Kontsevich integral, J. Math. Sci. 134 (2) (2006) 2017-2017, Table 1.
Index entries for sequences related to knots

Cf. A007473, A007478.

Cf. A014596.

Broadhurst gives a conjectured g.f.

Description corrected by Sergei Duzhin, Aug 29 2000

A007473 Dimension of space of Vassiliev knot invariants of order n.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 33, 60, 104, 184, 316, 548

Offset: 0

N. J. A. Sloane

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

Dror Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
Dror Bar-Natan, Bibliography of Vassiliev Invariants
D. J. Broadhurst, Conjectured enumeration of Vassiliev invariants, arXiv:q-alg/9709031, 1997.
Maksim Karev, On the primitive subspace of Lando framed graph bialgebra, arXiv:2307.00468 [math.CO], 2023.
Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12, arXiv:q-alg/9706022, 1997.
Eric Weisstein's World of Mathematics, Vassiliev Invariant.
Index entries for sequences related to knots

Cf. A007293 (first differences), A007478, A014595 (conjectured continuation).

G.f.: Product_{ m >= 1 } (1-y^m)^(-A007478(m)). - Andrey Zabolotskiy, Sep 19 2017

Broadhurst gives a conjectured explicit g.f. (different from A014595).

Description corrected by Sergei Duzhin, Aug 29 2000

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

A014605 Partial sums of A001935; at one time this was conjectured to agree with A007478.

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A007293 Dimension of space of weight systems of chord diagrams.

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A007473 Dimension of space of Vassiliev knot invariants of order n.

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

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