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A171108 a(n) is the Severi degree for curves of degree n and cogenus 2. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A135120 A027812 A171112 * A219439 A219140 A231513

Adjacent sequences:  A171105 A171106 A171107 * A171109 A171110 A171111

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 27 2010

EXTENSIONS

New name from Andrey Zabolotskiy, Jan 18 2021

STATUS

approved

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Last modified September 21 06:18 EDT 2021. Contains 347596 sequences. (Running on oeis4.)