A171116 -id:A171116 - cp's OEIS frontend

A171108 a(n) is the Severi degree for curves of degree n and cogenus 2.

0, 0, 21, 225, 882, 2370, 5175, 9891, 17220, 27972, 43065, 63525, 90486, 125190, 168987, 223335, 289800, 370056, 465885, 579177, 711930, 866250, 1044351, 1248555, 1481292, 1745100, 2042625, 2376621, 2749950, 3165582, 3626595, 4136175, 4697616, 5314320

Offset: 1

N. J. A. Sloane, Sep 27 2010

Severi degree N(n, delta) is the number of degree n plane curves which have delta nodes and pass through a generic configuration of n*(n+3)/2-delta points on the plane. delta is called the cogenus of these curves. See Fomin and Mikhalkin (2010), Section 1.2 "Combinatorial rules for Gromov-Witten invariants and Severi degrees" and 5 "Node polynomials". - Andrey Zabolotskiy, Jan 18 2021

Colin Barker, Table of n, a(n) for n = 1..1000
Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.
Florian Block, Susan Jane Colley, and Gary Kennedy, Computing Severi degrees with long-edge graphs, Bulletin of the Brazilian Mathematical Society, New Series 45.4 (2014): 625-647. Also arXiv:1303.5308 [math.AG], 2013 (see first page).
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Severi degrees N(n, delta) for other values of delta: A033428(n-1) (1), A171113 (3), A328551 (4), A328552 (5), A171116 (6).

Table[3(n-1)(n-2)(3n^2-3n-11)/2,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,0,21,225,882},40] (* Harvey P. Dale, Feb 01 2013 *)

concat([0,0], Vec(3*x^3*(7 + 40*x - 11*x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Nov 01 2019

a(n) = 3*(n-1)*(n-2)*(3*n^2-3*n-11)/2.
a(1)=0, a(2)=0, a(3)=21, a(4)=225, a(5)=882, a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Feb 01 2013
G.f.: 3*x^2*(-7-40*x+11*x^2) / (x-1)^5 . - R. J. Mathar, Dec 19 2013

New name from Andrey Zabolotskiy, Jan 18 2021

A171105 Multicomponent Gromov-Witten invariants for genus 0.

Original entry on oeis.org

1, 1, 12, 675, 109781, 40047888

Offset: 1

N. J. A. Sloane, Sep 27 2010

In this entry and in A171104, a multicomponent Gromov-Witten invariant is the number of (possibly reducible, hence "multicomponent") curves in CP^2 of degree n and genus g passing through given 3n-1+g points, so this is the Severi degree N(n, delta) where cogenus delta = (n-1)*(n-2)/2 - g, cf. A171108 and references therein. In particular, a(5) = A171116(5). - Andrey Zabolotskiy, May 04 2022

Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643. See Appendix B.
Grigory Mikhalkin, Enumerative tropical algebraic geometry in R^2, arXiv:math/0312530 [math.AG], 2003-2004.

Cf. A171104, A171108, A171116.

Cf. Gromov-Witten invariants, counting irreducible curves only: A171109, A171110, A171111.

a(5)-a(6) added by Andrey Zabolotskiy, May 04 2022

cp's OEIS Frontend

A171108 a(n) is the Severi degree for curves of degree n and cogenus 2.

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