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A014595 Conjectured dimensions of spaces of weight systems of chord diagrams. 2

%I

%S 1,1,2,3,6,10,19,33,60,104,184,316,548,931,1588,2676,4511,7539,12590,

%T 20890,34603,57036,93804,153655,251109,408961,664467,1076398,1739660,

%U 2804166,4510035,7236242,11585908,18509442,29511312,46957178,74575323,118213424,187052097,295453415

%N Conjectured dimensions of spaces of weight systems of chord diagrams.

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

%H D. Bar-Natan, <a href="https://doi.org/10.1016/0040-9383(95)93237-2">On the Vassiliev Knot Invariants</a>, Topology 34 (1995) 423-472.

%H D. Bar-Natan, <a href="http://www.pdmi.ras.ru/~duzhin/VasBib/">Bibliography of Vassiliev Invariants</a>.

%H Birman, Joan S., <a href="https://doi.org/10.1090/S0273-0979-1993-00389-6">New points of view in knot theory</a>, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253-287.

%H D. J. Broadhurst, <a href="https://arxiv.org/abs/q-alg/9709031"> Conjectured enumeration of Vassiliev invariants</a>, arXiv:q-alg/9709031, 1997 [different conjectured values].

%H Jan Kneissler, <a href="https://arxiv.org/abs/q-alg/9706022">The number of primitive Vassiliev invariants of degree up to 12</a>, arXiv:q-alg/9706022, 1997.

%H <a href="/index/K#knots">Index entries for sequences related to knots</a>

%F G.f.: Product_{ m >= 1 } (1-y^m)^(-A014605(m)). - _Andrey Zabolotskiy_, Sep 15 2017

%t terms = 40; QP = QPochhammer; A014605 = Join[{1, 0, 0, 0}, CoefficientList[ QP[q^4]/QP[q] + O[q]^terms, q]] // Accumulate;

%t gf = Product[(1 - y^m)^(-A014605[[m+1]]), {m, 1, terms}] + O[y]^terms;

%t CoefficientList[gf, y] (* _Jean-Fran├žois Alcover_, Jul 21 2018 *)

%Y 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.

%K nonn

%O 0,3

%A _David Broadhurst_

%E More terms from _Joerg Arndt_, Sep 19 2017

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Last modified January 25 23:08 EST 2020. Contains 331270 sequences. (Running on oeis4.)