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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 27 2010
EXTENSIONS
New name from Andrey Zabolotskiy, Jan 18 2021
STATUS
approved