A067640 -id:A067640 - cp's OEIS frontend

2, 20, 210, 2352, 27720, 339768, 4294290, 55621280, 734959368, 9873696560, 134510127752, 1854385377600, 25828939188000, 362995937665200, 5141806953167250, 73343003232628800, 1052697272275341000, 15194039267330154000, 220410039466873456200

Offset: 0

N. J. A. Sloane, Feb 05 2002

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots, arXiv:math-ph/0102015, 2001-2002.

Cf. A005568 (row 0), A067637 (row 2), A067638 (row 3), A067639 (row 4).

seq((2*n+2)!*(2*n+4)!/(n!*((n+2)!)^2*(n+3)!),n=0..30); # James Sellers, Feb 11 2002; adapted to offset 0 by Georg Fischer, May 29 2021

RecurrenceTable[{n*(n+2)*(n+3)*a[n] - 4*(n+1)*(2*n+1)*(2*n+3)*a[n-1] == 0, a[0]==2},a,{n,0,16}] (* Georg Fischer, May 29 2021 *)

a(n) = (2*n+2)!*(2*n+4)!/(n!*((n+2)!)^2*(n+3)!). [adapted to offset 0 by Georg Fischer, May 29 2021]
D-finite with recurrence: a(0) = 2, n*(n+2)*(n+3)*a(n) - 4*(n+1)*(2*n+1)*(2*n+3)*a(n-1) = 0 for n >= 1. - Georg Fischer, May 29 2021
a(n) ~ 2^(4*n + 6) / (Pi*n^2). - Vaclav Kotesovec, May 29 2021

More terms from James Sellers, Feb 11 2002

A067641 Column 1 of table in A067640.

Original entry on oeis.org

2, 20, 174, 1504, 13300, 120744, 1122198, 10638464, 102541428, 1002305040, 9914663308, 99085515840, 999104604784, 10153152363648, 103892246982390, 1069610792999424, 11072575568623300, 115189593628215600

Offset: 0

N. J. A. Sloane, Feb 05 2002

nonn

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots

A067642 Column 2 of table in A067640.

Original entry on oeis.org

10, 210, 2992, 37100, 433620, 4928798, 55237824, 614451348, 6807871480, 75275707584, 831595048320, 9185000522880, 101470031154352, 1121497913694390, 12403035430713344, 137266650274351716

Offset: 0

N. J. A. Sloane, Feb 05 2002

nonn

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots

A067643 Column 3 of table in A067640.

Original entry on oeis.org

70, 2352, 47820, 784672, 11515714, 158295072, 2086803540, 26737722400, 335676172480, 4150940757440, 50739269522864, 614607881444256, 7390867767651290

Offset: 0

N. J. A. Sloane, Feb 05 2002

nonn

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots

A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.

Original entry on oeis.org

1, 2, 8, 42, 260, 1796, 13396, 105706, 870772, 7420836, 65004584, 582521748, 5320936416, 49402687392, 465189744448, 4434492302426, 42731740126228, 415736458808868, 4079436831493480, 40338413922226212, 401652846850965808, 4024556509468827432, 40558226664529024000, 410887438338905738908, 4182776248940752113344, 42770152711524569532616, 439143340987014152920384, 4526179842103708969039296

Offset: 0

Sergei Duzhin, Nov 11 2000

Also the number of knot diagrams with n crossings and two outgoing strings.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
S. M. Gusein-Zade, On the enumeration of curves from infinity to infinity, in: Singularities and Bifurcations, Adv. Sov. Math., v. 21 (1994), pp. 189-198.

Steven R. Finch, Knots, links and tangles, August 8, 2003. [Cached copy, with permission of the author]
S. M. Gusein-Zade and F. S. Duzhin, On the number of topological types of plane curves; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. English translation: Russian Mathematical Surveys 53 (1998) 626-627. Related program and data.
J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots
J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Colored Links, J. Knot Theory, 10 (2001), 1233-1267.
Christoph Lamm, The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots, arXiv:2410.06601 [math.GT], 2024. See p. 14.
P. Zinn-Justin and J.-B. Zuber, Knot theory and matrix integrals, arXiv:1006.1812 [math-ph], 2010.
Index entries for sequences related to knots

Cf. A008980, A008981, A008982, A008983, A008984, A008985.
A151374 enumerates the long curves having Gauss diagrams without intersections, cf. A118814.
Cf. A067647, A067648.
A column of the triangles in A067640 and A062038.

Extended to n = 22 by J. L. Jacobsen and Paul Zinn-Justin, Jan 30 2002

More terms from Paul Zinn-Justin, Dec 13 2016

cp's OEIS Frontend

A067637 Row 2 of table in A067640.

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A067638 Row 3 of table in A067640.

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A067639 Row 4 of table in A067640.

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A067636 Row 1 of table in A067640.

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A067641 Column 1 of table in A067640.

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A067642 Column 2 of table in A067640.

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A067643 Column 3 of table in A067640.

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A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.

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